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CO2 is described as a supercritical gas when the pressure is above the critical pressure and the temperature is above the critical temperature. When the pressure is above the critical pressure while the temperature is below the critical temperature, it is simply described as a liquid. For CO2 the critical pressure is equal to 7377 kPa, and the critical temperature 31.1°C. The density at the critical point is 481 kg/m3. Supercritical CO2 is non-explosive, non-flammable, nontoxic, thermally stable, and readily available at low cost. It also has fluid property characteristics that make it attractive for advanced power plants. It has a relatively low critical pressure (7377 kPa versus 22064 kPa for steam). This means that it can be kept in a supercritical gas state throughout the cycle (as opposed to water/steam that requires boiling and condensation), thereby reducing the complexity of the system. Since the lowest temperature in the Brayton cycle is kept just above the critical temperature of 31.1°C, the cooling can done with atmospheric air. This is a significant advantage in regions where the solar power potential is high but cooling via cooling towers is not possible due to water scarcity. The temperature-entropy and temperature-density diagrams for CO2 are shown below. The two-phase region is indicated by the area enclosed below the black line and the critical point is indicated by the black dot. 500 500 b. 400 400 300 300 Temperature (EC) Temperature (°C] 200 200 100 100 0.5 10 1.5 20 2.5 3.0 3.5 200 400 600 800 1000 Entropy (kJ/kg/K] Density [kg/m^3] Consider the behaviour at three different constant pressure levels: at a low pressure of 100 kPa (green line), at the critical pressure of 7377 kPa (red line) and at a high pressure of 25000 kPa (blue line). On the large T-p diagram the green low pressure line is not visible since it lies very close to the y-axis. The inserted smaller T-p graph shows a closeup of the area right next to the y-axis, in which the green line is visible. On the T-s diagram in the area far away from the two-phase region, i.e. to the right of the dashed line, the trends for all the pressures are similar corresponding to ideal gas behaviour. This is because of the relatively low density in that region, which can be seen on the T-p diagram, where the corresponding area is to the left of the dashed line. For an ideal gas the entropy and density depend on both the 2 N
2- temperature and pressure, and (for constant cp) the entropy change is given by T, =c, In - R In 22, while the density can be found using the ideal gas law $, - s = T P. pa Closer to the two-phase region the behaviour is more complex. For RT instance, at the critical pressure and around the critical temperature the density changes very quickly from around 260 kg/m3 at 35°C to 760 kg/m3 at 25°C. This means that the density increases by 200% within a 10°C temperature range. The density of SCO2 is quite high, even at high temperatures. For instance, at 25000 kPa and 545°C the density is 155 kg/m², while for air it is 97 kg/m2 and for steam 79 kg/m3. This implies that for the same mass flow rates the sizes of piping and turbo machines will be significantly smaller for CO2 than for air or for steam, and therefore require less capital cost. Consider the heat capacity and thermal conductivity behaviour shown below. On both the T-Cp and T-k diagrams the ideal gas region is to the left of the dashed line. On the T-Cp diagram the saturated vapour line folds back almost on top of the saturated liquid line with the two-phase region below the black lines. 500 500 400 400 300 300 Temperature [°C] Temperature (°C] 200 200 100 100 0 8 10 12 0.02 0.04 0.06 0.08 0.10 0.12 0.14 6.60 Heat capacity (kJ/kg] Conductivity (W/mK] In the ideal gas region, the heat capacity is only weakly dependent on pressure, and when the temperature gets high enough it is almost independent of pressure. This is why for an ideal gas the assumption is usually that co= f(T). At the critical pressure and near the critical temperature both the heat capacity and the conductivity increase drastically and tends to infinity at the critical point. This clearly shows that around the critical point one cannot assume ideal gas behaviour, and this needs to be considered when analysing CO2 power cycle performance. Finally, near its critical point sCO2 becomes more incompressible, and hence, the compression work can be decreased substantially, leading to high cycle efficiency.
You are required to provide input to the conceptual design phase of the sCO2 Brayton cycle. For this purpose, you will develop a Python program that you can use to do parametric studies of the power cycle performance for different scenarios. Before embarking on the development of the Python code, you will do some manual calculations to get a sense of the cycle performance at the proposed nominal operating conditions. The results from this calculation will also be used to verify the Python code before you use it to generate results for the parametric study. The proposed nominal operating point for the Brayton cycle is as follows: P. = 7600 kPa, PR = 3.2895, T, = 60°C.T. = 545°C and W = 40 MW. The nominal performance characteristics of the different components are: nc =0.89, n. = 0.93, E, = 0.85, Erc = 0.85, Exx = 0.95 and the pressure drop through each of the heat exchangers is 2% of the pressure at its inlet. The mechanical efficiency of the shaft is 1m = 0.99 and of the generator ng =0.9. Provide your solutions for Q1, Q2, Q3 in a single pdf file, and Q4 in a second pdf file. Q1 26 marks Do a manual calculation of the cycle performance at the proposed nominal operating point, based on the simplifying assumption that the CO2 is an ideal gas with constant values of c, =0.846 kJ/kgk and y =1.289. Show all the calculation steps. (0) Draw the T-s and p-h schematic diagrams. () Determine the pressure and temperature at each point in the cycle. (ii) Determine the required gas mass flow rate through the cycle. (iv) Determine the power input and output of each of the machines as well as the rate of heat transfer in each of the heat exchangers. (v) Determine the cycle thermal efficiency, the overall plant efficiency, as well as the cycle thermal efficiency if an ideal heat engine is operated between the same maximum and minimum temperatures. (vi) Prove that you have obtained an overall energy balance for the cycle. 24 marks Repeat the manual calculation to analyse the cycle performance at the proposed nominal operating point, but this time using the CO2 real gas properties provided by CoolProp using the Excel add-in. Show all the calculation steps. (0) Determine the pressure, enthalpy, and temperature at each point in the cycle. () Determine the required gas mass flow rate through the cycle. (ii) Determine the power input and output of each of the machines as well as the rate of heat transfer in each of the heat exchangers. (iv) Determine the cycle thermal efficiency, the overall plant efficiency, as well as the cycle thermal efficiency if an ideal heat engine is operated between the same maximum and minimum temperatures. (v) Prove that you have obtained an overall energy balance for the cycle. Q2
Q3 6 marks Now develop a Python program that uses the CoolProp module to repeat the analysis done in Q2. In order to verify your implementation, first compare the results obtained for Q2 with the Python program with that of your manual calculation. Now use your Python program to do a parametric study where you keep all the cycle parameters at the nominal values provided above, except for the pressure ratio of the compressor. Investigate the impact of varying the pressure ratio between 1.2 and 8.0. As the output to this exercise provide the following six graphs: 1. Cycle thermal efficiency versus the compressor pressure ratio. 2. Mass flow rate versus the compressor pressure ratio. 3. Compressor power versus the compressor pressure ratio. 4. Turbine power versus the compressor pressure ratio. 5. Precooler heat transfer rate versus the compressor pressure ratio. 6. Heater heat transfer rate versus the compressor pressure ratio. Provide your solutions for Q1, Q2 and Q3 in a single pdf file. 10 marks Provide a printout of your Python program together with its results for the analysis in Q2 in a separate pdf file. Marks will be given for using the "CyclePlots.py" module to correctly plot the T-s, p-h, p-v and h-s diagrams for the operating point in Q2 as part of your program output. Q4 5