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wWe Wc (7) m Cycle Efficiency: Qin Heat Exchanger Effectiveness: h5 h6 h3 h2 3. Mathematical Modeling 3.1. Cycle Analysis and Optimization We Wc (8) In order to determine the most efficient operating point, a parametric analysis is performed using first order thermodynamics. For the prototype cycle, limits are chosen to be 35°C and pressure above 7.4 MPa to maintain the fluid above the critical point and avoid two-phase flow. Temperature on the high side is dictated by input from a low quality heat source, which generally originates at 200 °C - 500 °C [26]. Pump and expander efficiencies are determined from assumptions outlined in section 4. The remaining fixed parameters are determined based on system requirements of recovering energy from low-quality heat using SCO2. The three degrees of freedom considered for the parametric analysis are: volumetric expansion ratio, high side pressure, and the temperature drop across the cooler. In order to close the thermodynamic cycle the temperature drop across the cooler was assumed to be a function of all other parameters. A MATLAB program is used to vary the three degrees of freedom, discard any physically impossible cycles, then determine the most efficient parameter combination. The flow chart for this program is shown in Figure 2. In order to eliminate two degrees of freedom, the Carnot efficiency ηcarnot as given in Eqn. (10) dictates that the greatest efficiency will be achieved when Tcold is minimized and Thot is maximized. Therefore, maxima and minima for these values are selected where Tcold is above the supercritical region and Thot is lower than the waste heat source. carnot 1Tcold (10) Thot Figure 2. This flow chart depicts the optimization program used to determine the most efficiency cycle. Pressure index varies from 14 MPa to 20 MPa with the upper bound of nP at 20 MPa. Similarly, the Volumetric Expansion Index varies from 1.1 to 2.45 with the upper bound of nV at 2.45. Figure 3 is generated using the same MATLAB code and shows the dynamic relationship dictated by the governing equations and boundary conditions (i.e. high-side and low-side pressure, and state-points from the p-h diagram). Using Figure 3 the required temperature drop across the heat exchanger, cooler, and cycle efficiency may be determined for a given heat exchanger effectiveness. Figure 3 can thus be viewed as a road-map in SCO2 waste heat recovery cycle component design and selection. Using MATLAB on a 64-bit workstation the one source of possible numerical error is round-off error. Herein round-off error is estimated to be 10%, which is bounded by the at least 15~20% uncertainty associated with the SCO2 thermo-physical properties as obtained from REFPROPS [10]. After our design optimization was concluded, the state points shown in Table 1 were recorded to document the optimized SCO2 Rankine Regenerative Cycle. Table 1 is an itemized tabulation of where the state points fall on the p-h and T-s phase diagrams for (9) 4PDF Image | Waste Heat Energy Supercritical Carbon Dioxide Recovery Cycle Analysis and Design
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