Turbomachinery Design Project M1337 Laser Tank “Yancy” Rob Machen Griffin Pafford 1 | P a g e Background The M1337 Laser Tank is also known as the Yancy, named for Major General Philip Yancy, former commander of the 7 th Armored Brigade. It is a turbine powered, semiautonomous tank designed for both anti ‐ air and missile defense. It is equipped with a Bahamut 1 ‐ Megawatt laser system, produced by General Atomic. The Yancy is built upon the Abrams platform, equipped with a powerful enough turbine engine to power the 1 Megawatt laser that is the heart of the system. When the laser is not in use, the M1337 can travel up to 60 mph over level ground, though it is limited to 45 mph for safety concerns. The axial flow type turbine used in the Abrams platform is also well suited for the M1337 Laser Tank, even with the power demand mandated by the 1 Megawatt laser. The original Honeywell AGT 1500 produced a peak output of 1500 hp, more than enough to power the 1 Megawatt (or 1341 hp) laser to function at its rated strength. However, due to generator and motor losses in the M1337, this is not sufficient to meet the needs of the laser. Therefore the more powerful Honeywell AGT 1850, with a peak power 1850 hp, is necessary to replace the previous turbine. This engine should be more than enough to overcome the losses in the system. 2 | P a g e Ideal Cycle Considerations For simplicity, it has been assumed that the power generation cycle used for the M1337 Laser Tank is an ideal Brayton Regeneration (Heat ‐ Exchange) Cycle. The remaining question is whether or not to use reheat in the cycle. A Brayton Regeneration cycle without reheat is shown below in Figure 1 and a cycle with reheat is shown in Figure 2. Figure 1. Brayton Cycle with Regeneration [1] Figure 2. Brayton Cycle with Regeneration and Reheat [1] It is known that the efficiency of a Brayton Cycle engine with regeneration but without reheat is less efficient than with reheat, as shown by the graph in Figure 3 below (the efficiency of a simple Brayton Cycle is also included for comparison). The same is true with power generated by the engine, as shown in Figure 4 as a unit ‐ less quantity. (The equations used for each of the graphs in these figures can be found in Appendix A.) One might think this would make the answer to the question “reheat or no reheat” incredibly obvious. However, adding reheat also increases the amount of equipment in the cycle, and therefore the cost, weight, and required space of the engine. Weight and space are especially important when designing for a vehicle application. Since the vehicle in question is a tank that is already destined to be fairly large and weigh around 60 tons, though, weight and space do not seem as critical. With these considerations in mind, a Brayton Regeneration Cycle with reheat seems to be the best choice. 3 | P a g e 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 Pressure Ratio Efficiency reheat η r 2 , ( ) reheat η r 3 , ( ) reheat η r 4 , ( ) reheat η r 5 , ( ) simple r ( ) heatexchange η r 2 , ( ) heatexchange η r 3 , ( ) heatexchange η r 4 , ( ) heatexchange η r 5 , ( ) r Figure 3. Brayton Cycle Thermal Efficiency With & Without Reheat 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 Pressure Ratio Power' reheatP r 2 , ( ) reheatP r 3 , ( ) reheatP r 4 , ( ) reheatP r 5 , ( ) heatexchange P r 2 , ( ) heatexchange P r 3 , ( ) heatexchange P r 4 , ( ) heatexchange P r 5 , ( ) r Figure 4. Brayton Cycle Power With & Without Reheat 4 | P a g e Note that r is the cycle pressure ratio and t is the cycle temperature ratio. Both Figure 3 and Figure 4 show that higher temperature ratios provide more power, which is important for the M1337 Laser Tank’s requirement of 1850 hp (1.38 MW). However, the turbines have been limited in such a way as to only allow a maximum inlet temperature of 1100K. With an ambient temperature of 298K, this means that the highest feasible temperature ratio is 3.691 for this vehicle. Furthermore, for heat exchange with reheat to be effective (i.e. more efficient than a simple cycle), the pressure ratio r must remain below the simple cycle threshold. Therefore, the highest pressure ratio should be limited to about 13 for a cycle with a 3.691 temperature ratio, as can be seen in Figure 5 below (equation provided in Appendix A). Otherwise, it becomes more efficient to use a simple cycle. 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Pressure Ratio Efficiency vs Power' reheat η r 3.691 , ( ) reheatP r 3.691 , ( ) simple r ( ) r Figure 5. Brayton Regeneration Cycle with Reheat and t = 3.691 In conclusion, judging by these ideal cycle considerations, it is clear that the best cycle for the M1337 Laser Tank would be a Brayton Regeneration Cycle with reheat and a temperature ratio of 3.691. The pressure ratio for the ideal cycle varies from 1 to about 13, with a higher pressure ratio providing more power but less efficiency. With a pressure ratio larger than 13, a simple cycle may be more appropriate. 5 | P a g e Real Cycle Considerations Calculations for a realistic cycle must include estimation for energy losses due to irreversibilities of real components. Each component of the cycle has an efficiency, effectiveness, or pressure loss. Given component efficiencies are provided in Table 1 below, while pressure losses are provided in Table 2. Table 1. Efficiencies Efficiencies Symbol Value Compressor Polytropic η ∞ c 0.85 Turbine Polytropic η ∞ t 0.85 Mechanical Transmission η m 0.99 Combustion η b 0.98 Heat Exchanger Effectiveness ε 0.80 Table 2. Power Losses Pressure Losses Symbol Value Combustion Chamber Δ P b 6% Heat Exchange Air Side Δ P ha 5% Heat Exchange Gas Side Δ P hg 5% The highest possible temperate ratio, t , is already known from the ideal cycle considerations to be 3.691. However, the best pressure ratio, r, must also be calculated while taking the above mentioned efficiencies and losses into consideration. The thermal efficiency of the cycle is heavily affected by the heat exchanger effectiveness, as shown below in Figure 6 (for a temperature ratio of 3.691; graph equations are provided in Appendix A). In this figure, a Brayton Cycle with heat ‐ exchange and reheat is compared to a cycle with only heat ‐ exchange and to a simple cycle, much as in Figure 3. Note that the thermal efficiency of the cycle with only heat ‐ exchange is much lower than even the simple cycle, now that heat transfer effectiveness is taken into account. This is further proof that a Brayton Regeneration Cycle without reheat is not a good choice for this application, especially since the platform should have plenty of room to accommodate reheating. Also note that the simple cycle threshold, when the Brayton Regeneration Cycle with reheat become less efficient becomes less efficient than the simple cycle, has dropped from about 13 to 12. 6 | P a g e 0 2 4 6 8 10 12 14 16 0 0.086 0.171 0.257 0.343 0.429 0.514 0.6 Pressure Ratio Efficiency (with HE eff.) RH η r 3.691 , ( ) HE η r 3.691 , ( ) simple r ( ) r Figure 6. Brayton Regeneration Cycle With/Without Reheat, t = 3.691, and 0.80 H.E. Effectiveness Most importantly, Figure 6 shows that the highest thermal efficiency for the Brayton Regeneration Cycle with reheat, 52.43%, occurs when the pressure ratio, r , is 6.15. Using this pressure ratio and considering the efficiencies and losses given above, one can also find the specific work output, specific fuel ratio, and cycle efficiency (calculations provided in Appendix B). Note that the fluid traveling through the compressor is air considered as an ideal gas, with a specific heat ratio, γ a , of 1.4, a specific heat, c pa , of 1.005 kJ/kg/K, and a density, ρ , of 1.227 kg/m 3 The fluid traveling through both turbines is a mixture of fuel and air with a specific heat ratio, γ g , of 1.333 and a specific heat, c pg , of 1.148 kJ/kg/K. The fuel in this application is kerosene burned in air ( Δ H 25 = 43,100 kJ/kg). Under these conditions, the calculated work ‐ per ‐ unit ‐ mass ‐ of ‐ air generated by the compressor and turbines are provided in Table 3 below. Note that work from the compressor is negative, since it requires energy to compress the fluid. Also, the work generated by the low pressure turbine is equivalent to that of the high pressure turbine. Table 3. Component Work ‐ per ‐ Unit ‐ Mass ‐ of ‐ Air Component Symbol Work/Mass (kJ/kg) Compressor ‐ω tc ‐ 254.562 High Pressure Turbine ω HPT 195.784 Low Pressure Turbine ω LPT 195.784 Total (Net) ω net 137.006 7 | P a g e The specific fuel ratios for the two combustors in the cycle are provided in Table 4. Table 4. Specific Fuel Ratios for Combustors [2] Component Symbol Fuel Ratio Combustor 1 f 1 0.007041 Combustor 2 f 2 0.005816 Total ftotal 0.012857 Dividing the total fuel ratio by the net work ‐ per ‐ unit ‐ mass provides the specific fuel consumption (SFC) of this cycle, 9.384 x 10 ‐ 5 kilograms of fuel per kilojoule of energy generated (kg/kJ), or in more common power units, SFC = 0.338 kg/kWh. Finally, using the SFC and the enthalpy of reaction for kerosene at a reference temperature of 25°C, the cycle efficiency, η cycle , is found to be 26.97%. Furthermore, the new thermal efficiency for the cycle, calculated by dividing the net work ‐ per ‐ unit ‐ mass by the heat ‐ transfer from the combustors, is 30.88%. (All real cycle calculations are provided in Appendix B) To confirm that a pressure ratio of 6.15 is actually the optimal pressure ratio, the net work (with and without units), specific fuel consumption, cycle efficiency, and thermal efficiency values were all recalculated for a variety of pressure ratios from 5 to 12. Pressure ratios lower than 5 could not be used, as ideal fuel/air ratios are unavailable below this point. Pressure ratios above 12 are useless as a simple cycle becomes more efficient, as shown in Figure 6 above. The results of these calculations are displayed in Table 5 and Figure 7 below. Table 5. Differences by Pressure Ratios r (untiless) ω net (kJ/kg) ω ’ (unitless) SFC (kg/kWh) η cycle (%) η th (%) 5 127.03 0.4242 0.312 0.2674 0.3092 6 136.009 0.4541 0.311 0.2689 0.3092 6.15 137.006 0.4575 0.31 0.2697 0.3088 7 141.361 0.472 0.314 0.2656 0.3049 8 144.28 0.4818 0.318 0.2624 0.2984 9 145.493 0.4858 0.328 0.2545 0.2907 10 145.466 0.4857 0.338 0.2468 0.2822 11 144.516 0.4825 0.348 0.2399 0.2734 12 142.862 0.477 0.357 0.2337 0.2645 As shown in the above table, a pressure ratio of 6.15 really is the best ratio in terms of cycle efficiency. The most power can be generated with a pressure ratio of about 9, but SFC and efficiency suffer greatly. The highest thermal efficiency actually occurs with a pressure ratio slightly below 6, but less power is generated. 8 | P a g e Figure 7. Differences by Pressure Ratios With the optimal pressure ratio confirmed and the generated net work determined, a mass flow rate can also be found by dividing the desired power output of 1.38 MW by the work ‐ per ‐ unit ‐ mass ‐ of ‐ air. This comes out to a required mass ‐ airflow rate of 10.069 kg/s. Dividing this number by the density of air provide the volumetric airflow rate, 8.206 m 3 Assuming an air ‐ speed of 10 mph at the compressor inlet, the compressor must have an inlet area of 1.836 m 2 , or a diameter of 1.529 meters, in order for the turbine engine to get the air it needs to produce about 1.38 MW of power. Compressor Design The most crucial piece of information in designing the compressor for the M1337 Yancy was the mass flow rate of 10.0069 kg/m^3 obtained in the real cycle thermodynamic analysis. The analysis of the designed cycle determined that this mass flow rate was necessary to generate the 1.38 MW required by the Yancy. Several other simplifying assumptions were also made for the compressor design, including that the air moving through the compressor is an ideal gas and that the compressor experiences ideal intake flow. The compressor must also operate under subsonic flow, so the Mach number must be equal to or less than one. This allows that calculation of a maximum tip speed just under 350 m/s, which was then further reduced by 5% to ensure the air flow remains subsonic. Furthermore, the initial blade root ‐ to ‐ tip ratio is set at 0.8, meaning the height of the blade is one fifth the radius of the blade tip from the central axis. The mean radius of the annulus was selected to remain constant throughout the design. The axial flow through the compressor is assumed to be constant. Finally, a polytropic efficiency of 85% was given for the compressor. 9 | P a g e Figure 8. Example Compressor Blades To address concerns over the small difference in blade angles, the axial flow velocity was adjusted to increase the change in blade angles of the compressor’s first stage. The changes in the annulus mean radius, blade and stator heights, shaft rpm, blade angles and stagnation temperature according to the changes in axial flow velocity are displayed in Table 6 below (calculations provided in Appendix C). An axial velocity of 70 mph was chosen, as it provides a reasonable change in blade angle for a reasonable inlet velocity. A fan with a screen would be required in front of the compressor to provide this velocity and protect the engine from debris. Table 6. Various Specifications at Different Axial Flow Velocities Ca (mph) rm (ft) h1 (in) h2 (in) w (rpm) β 1 β 2 (Deg) Δβ ∆ Tos (K) 15 3.116 8.311 7.288 2973.541 0.505 0 23.903 20 2.699 7.198 6.312 3432.697 0.674 0.169 23.906 30 2.204 5.878 5.155 4201.194 1.011 0.337 23.914 40 1.91 5.092 4.465 4846.298 1.35 0.339 23.927 50 1.709 4.556 3.995 5411.392 1.69 0.34 23.942 60 1.56 4.161 3.648 5918.598 2.031 0.341 23.962 70 1.445 3.855 3.379 6380.964 2.374 0.343 23.984 80 1.353 3.608 3.162 6806.914 2.72 0.346 24.011 90 1.276 3.404 2.983 7202.242 3.068 0.348 24.041 10 | P a g e Once the axial flow velocity was chosen, the number of required compressor stages was estimated to be 12.345, rounding up to 13 stages, with an increase in stagnation temperature of 250.762 K. This is the appropriate change in temperature to provide the necessary pressure for the turbine cycle. The stage ‐ by ‐ stage calculations, provided in Table 7 below (calculations provided in Appendix C), showed this to be a poor estimate, as 13 stages would not provide this high an increase in stagnant temperature. Instead, 19 stages were required. Table 7. Compressor Design Blade Angles and Radii by Stage Stage Δ T (K) Λ β 1 (deg) β 2 (deg) α 2 (deg) α 3 (deg) rt2 (in) rr2 (in) rt3 (in) rr3 (in) 1 23.984 0.858 83.933 81.558 69.466 60.846 19.035 15.656 19.015 15.676 2 18.554 0.694 82.519 79.582 75.859 70.164 18.894 15.797 18.859 15.833 3 15.397 0.604 81.43 78.055 77.941 73.132 18.786 15.905 18.743 15.948 4 13.435 0.556 80.705 77.036 78.829 74.391 18.695 15.996 18.649 16.042 5 12.857 0.53 80.265 76.417 79.254 74.991 18.611 16.081 18.566 16.125 6 12.548 0.516 80.012 76.061 79.468 75.294 18.533 16.158 18.491 16.200 7 12.383 0.509 79.871 75.863 79.58 75.451 18.462 16.229 18.423 16.269 8 12.295 0.505 79.795 75.754 79.638 75.534 18.397 16.294 18.361 16.331 9 12.248 0.502 79.753 75.696 79.669 75.578 18.337 16.354 18.304 16.387 10 12.223 0.501 79.731 75.664 79.686 75.601 18.282 16.409 18.252 16.440 11 12.210 0.501 79.719 75.647 79.695 75.614 18.232 16.459 18.204 16.488 12 12.202 0.5 79.712 75.638 79.699 75.620 18.185 16.506 18.159 16.532 13 12.199 0.5 79.709 75.633 79.702 75.624 18.143 16.549 18.118 16.573 14 12.196 0.5 79.707 75.631 79.703 75.626 18.103 16.589 18.080 16.611 15 12.195 0.5 79.706 75.629 79.704 75.627 18.066 16.625 18.045 16.646 16 12.195 0.5 79.705 75.629 79.704 75.627 18.032 16.659 18.012 16.679 17 12.194 0.5 79.705 75.628 79.705 75.627 18.000 16.691 17.982 16.709 18 12.194 0.5 79.705 75.628 79.705 75.628 17.970 16.721 17.954 16.738 19 12.194 0.5 79.705 75.628 79.705 75.628 17.943 16.748 17.927 16.764 Total 255.703 The final compressor design can be deduced from the information presented in Table 6 and Table 7 above. The compressor is designed for a 70 mph air flow velocity at the inlet, the area of which is 0.872 ft 2 The inlet blade height is 3.855 inches, while the exit blade height is 1.163 inches. Assuming a height ‐ to ‐ chord ‐ length ratio of 3, the blade heights indicate that the total length of the compressor is roughly 25 inches. The compressor shaft rotates at about 6381 rpm, and the total stagnation temperature rise is 255.762 Kelvin, or 460.102 degrees Fahrenheit. The final pressure of the air at the compressor exit is about 644.29 kPa, and the final pressure ratio comes out to be 6.359. 11 | P a g e Normally, there are three primary concerns when designing a compressor. One is whether the maximum blade tip speed is faster than the speed of sound, which might result in stress issues. Another is the speed of the axial air flow, which should also remain less than the speed of sound (at or below Mach 1). If the Mach number of the axial flow is more than 1, the compressor would require a different, more complicated design. The last concern is excessive fluid deflection by the compressor blades, or high diffusion rates, which could result in blade stall. For this compressor design, all three of these concerns have been taken into account in the preliminary design. The maximum blade tip speed is only 732 mph, which is about 51 mph less than the speed of sound (roughly 783 mph). Furthermore, the Mach number of the axial flow is limited to 1, and the blade angles were designed in accordance with the de Haller number, 0.72, so as to limit diffusion to allowable levels. Turbine Design Designing the high and low pressure turbines of the cycle proved to be fairly simplistic in comparison to the compressor design. Since the pressure gradient is advantageous, there are no diffusion limitations and therefore fewer stages are necessary. Furthermore, the degree of reaction is 0.5 for all stages. No changes in blade angles are required once the blade angles are initially defined. The primary concern becomes the efficiency of the turbines. The preliminary design of both turbines depends on an inlet temperature of 1100 K, a constant axial flow velocity and a constant mean blade radius. The pressure delivered from the compressor and the predicted changes in temperature given by the real cycle turbine analysis (with the actual pressure ratio of 6.359 provided by the compressor; see Appendix C) are also required. The change in temperature across both the high and low pressure turbines comes out to be 173.836 K (see Appendix B). The low pressure turbine is also placed on the same axle as the high pressure turbine, so the rotational speed of both turbines match (6381 rpm). The number of stages for both turbines was set to 2, for a total of 4 stages. The mean radii of the turbine blades in both turbines were also chosen to be equivalent to the mean radius of the compressor, 1.445 ft, which also provided a median blade speed of about 659 mph (calculations provided in Appendix D). The temperature changes from the real cycle analysis were then used with the Smith Diagram (turbine loading/efficiency chart) to determine turbine efficiencies, 88%, and axial air flow velocity, 132 mph. The rotor and stator blade angles were also calculated in the process and are shown below in Table 8. Next, determining temperature, pressure, and density between the blades allowed the calculation of blade tip and root radii, and thereby the height of the blades. The inlet and exit blade radii and blade heights are also shown in Table 8 below. The exit area for the low pressure turbine comes out to be 3.902 ft 2 The final length of each is found to be 3.7 inches for the high pressure turbine and 3.1 inches for the low pressure turbine. 12 | P a g e Table 8. Turbine Design Blade Angles and Radii by Stage Stage β 2 β 3 α 1 α 2 rts1 (in) rrs1 (in) hs1 (in) rts3 (in) rrs3 (in) hs2 (in) 1H 17.935 79.361 17.935 79.361 17.940 16.751 1.189 18.145 16.751 1.394 2H 17.935 79.361 17.935 79.361 18.145 16.547 1.598 18.488 16.547 1.941 1L 17.935 79.361 17.935 79.361 18.737 15.955 2.782 19.215 15.955 3.260 2L 17.935 79.361 17.935 79.361 19.215 15.477 3.738 19.923 15.477 4.446 Two final checks were made on the high and low pressure turbine designs to ensure the designs were sound. First the flare angle of each turbine was measured. An angle greater than 25 degrees might allow exit flow separation, but both turbine flare angles were found to be about 5.8 degrees. Next, the mach number of the exit flow from both turbines was measured. As long as the mach numbers for the nozzle exit and leading edge of the rotor blades is less than 1.2 and 0.75, respectively, there should be no issues. The Mach numbers for the high and low pressure turbines, shown below in table 9, are clearly well below these limits, showing that both turbines are sound in design. Table 9. Turbine Air Flow Mach Numbers Mach # Nozzle exit Leading Rotor Blade Edge Limit < 1.2 < 0.75 HPT 0.558 0.108 LPT 0.603 0.117 Safety Issues and Practicality The M1337 ‘Yancy’ Laser Tank, shown below in Figure 9, is a very powerful machine powered by a Honeywell AGT 1850 fielding a Bahamut 1 megawatt class laser system. The electrical power that is supplied has to be stored in high density capacitors and if they were damaged they would probably explode fairly spectacularly destroying the vehicle. If a large object got past the screen and into the compressor, the whole engine would likely explode radially, throwing blades and other parts at tremendous velocity and effectively destroying the vehicle. If the Laser was damaged before firing or while firing it would likely explode destroying the vehicle. If the fuel tank were punctured and ignited it would likely explode and destroy the vehicle. However, due to the fact that the vehicle is semiautonomous with no onboard driver, these safety concerns do no bear a risk to an actual human being, so long is a safe distance from the tank is maintained. Also, since the M1337 would typically be deployed well behind friendly lines, it should not experience any serious attacks. If it does come under fire, its armor should be able to shrug off all small arms and most man portable rocket systems. The 13 | P a g e laser is very vulnerable, however, and should be treated with great care. In the event of traveling through hostile territory, the detachable armored casing for the laser should be put in place. The M1337 Laser Tank is a very versatile platform which can take down enemy drones, air craft, and even missiles when paired with its sister platform, the M87 Mobile Radar (Figure 10). The M1337’s 1 megawatt laser, shown in Figure 11 on page 15, is theoretically capable of melting through twenty feet of steel in one second, which is more than enough destructive power to take down anything in the sky today. The Laser turret is also capable of lowering its angle of attack below zero degrees to be able to fire at ground targets. In so doing, it would cut a terrible swath of destruction never before seen on the modern battlefield. 14 | P a g e Figure 9. M1337 'Yancy' Laser Tank ‐ Component Views 15 | P a g e Figure 10. Anti ‐ Missile Weapon Concept Figure 11. Bahamut 1 Megawatt Laser System 16 | P a g e Conclusions The final design of the M1337 Yancy’s turbine engine uses a Brayton Regeneration cycle with reheat. The compressor is 25 inches long, and the high and low pressure turbines are 3.7 and 3.1 inches long, respectively. The maximum diameter of the engine, not including the thickness of armor plating, is about 40 inches at the intake of the compressor as well as at the exit of the low pressure turbine. The engine is design for an air mass flow rate of 10.0069 kg/s, a pressure ratio of 6.395, and a rotational speed of about 6381 rpm. The compressor is designed for an axial flow velocity of about 70 mph with 19 stages and a stagnation temperature rise of 255.703 K. The compressor requires 2.572 MW of power. The turbines are designed for a maximum efficiency of 88% with 1100 K inlet temperatures and an airflow velocity of 132 mph. The high pressure turbine generates 1.998 MW while the low pressure turbine generates 2.017 MW for a total of 4.015 MW generated. The engine produces a total net power of 1.443 MW, just enough to power the M1337 ‘Yancy’ and cover any losses to efficiency. 17 | P a g e References [1] Bhattacharjee, Subrata. "Engineering Thermodynamics: Problems and Solutions, Chapter ‐ 8." TEST: Web Based Software for Thermodynamic Property Evaluation and Thermal Systems Analysis San Diego State University, 31 Oct. 2010. Web. 22 Mar. 2012. <http://thermo.sdsu.edu/testhome/Test/problems/chapter08/chapter08Local.html>. [2] Saravanamuttoo, Herb I. H., Gorden F. C. Rogers, Henry Cohen, and Paul V. Straznicky. Gas Turbine Theory Harlow, England: Pearson Prentice Hall, 2009. Print. [3] "Figure 2 ‐ 10. ‐ Zero Indexing the Compressor Rotor." Engine Mechanics Integrated Publishing. Web. 26 Apr. 2012. <http://enginemechanics.tpub.com/14111/css/14111_54.htm>. Appendix A: Efficiency and Power Equations and Figures Efficiency Specific Heat Ratio for Air: γ 1.4 := Define c: c r ( ) r γ 1 − γ := simple r ( ) 1 1 c r ( ) − := Thermal Efficiency: reheat η r t , ( ) 2t c r ( ) − 1 + 2t c r ( ) − 2t 2t c r ( ) − := r 1.01 1.02 , 16 .. := heatexchange η r t , ( ) 1 r γ 1 − γ t − := 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 Pressure Ratio Efficiency reheat η r 2 , ( ) reheat η r 3 , ( ) reheat η r 4 , ( ) reheat η r 5 , ( ) simple r ( ) heatexchange η r 2 , ( ) heatexchange η r 3 , ( ) heatexchange η r 4 , ( ) heatexchange η r 5 , ( ) r A1 Power reheatP r t , ( ) 2t c r ( ) − 1 + 2t c r ( ) − := heatexchangeP r t , ( ) 1 c r ( ) − t 1 1 c r ( ) ⎛ ⎜ ⎝ ⎞ ⎟ ⎠ − ⎡ ⎢ ⎣ ⎤ ⎥ ⎦ + := 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 Pressure Ratio Power' reheatP r 2 , ( ) reheatP r 3 , ( ) reheatP r 4 , ( ) reheatP r 5 , ( ) heatexchangeP r 2 , ( ) heatexchangeP r 3 , ( ) heatexchangeP r 4 , ( ) heatexchangeP r 5 , ( ) r Efficiency vs Power @ t = 3.691 0 2 4 6 8 10 12 14 16 0 0.2 0.4 0.6 0.8 1 1.2 1.4 Pressure Ratio Efficiency vs Power' reheat η r 3.691 , ( ) reheatP r 3.691 , ( ) simple r ( ) r A2