INFINITY TURBINE LLC We specialize in designs, plans, licensing, consulting, design services, and surplus spare parts. We no longer manufacture turbines or CO2 systems. More Info...
TEL: +1-608-238-6001 (Chicago Time Zone ) USA
Email: greg@infinityturbine.com
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Temperature Drop Through Multi-Stage Supercritical CO2 Turbines with Recuperation IntroductionIn a supercritical CO2 (sCO2) Brayton cycle, turbine expansion converts the thermal and pressure energy of supercritical CO2 into mechanical shaft work. During this expansion, the CO2 temperature drops as enthalpy is converted to work.Understanding how temperature decreases across single-stage and multi-stage turbines is essential for cycle design, especially when combined with recuperation, which recovers waste heat from the turbine exhaust to preheat the compressed CO2. The number of turbine stages, total pressure ratio, and degree of recuperation determine the exit temperature profile and overall cycle efficiency.The Thermodynamic FoundationThe turbine temperature drop is governed by the isentropic expansion relation:[T_{out,isen} = T_{in} times left( frac{P_{out}}{P_{in}} right)^{frac{(gamma • 1)}{gamma}}]where( T_{in} ) = turbine inlet temperature (K)( P_{in}, P_{out} ) = inlet and outlet pressures( gamma ) = specific heat ratio (~1.3 for CO2 in the supercritical region)The actual exit temperature is higher than the isentropic value due to turbine efficiency:[T_{out,actual} = T_{in} • eta_t times (T_{in} • T_{out,isen})]with typical sCO2 turbine isentropic efficiencies between 80–90%.Typical Conditions for AnalysisLet’s assume a representative sCO2 Brayton turbine setup:Turbine inlet temperature (TIT): 600°C (873 K)Turbine inlet pressure: 20 MPaTurbine outlet pressure (overall): 7 MPaEffective specific heat ratio (γ): 1.3Turbine efficiency: 85%Recuperation effectiveness: 85–90%The total pressure ratio (PR) = 20 / 7 ≈ 2.86.Single-Stage ExpansionA single-stage turbine handles the entire pressure ratio in one expansion.For PR = 2.86:[T_{out,isen} = 873 times (1/2.86)^{(0.3/1.3)} approx 873 times 0.773 = 675 K][T_{out,actual} = 873 • 0.85 times (873 • 675) = 873 • 0.85 times 198 = 704 K]Temperature drop: 873 − 704 = 169 K (about 169°C)Exit temperature: ~431°CThis high single-stage drop yields strong shaft power but also limits recuperator temperature difference, since the outlet temperature is still high.Two-Stage ExpansionWith two stages, each handles roughly half the total pressure ratio:[PR_{stage} = sqrt{2.86} = 1.69]Stage 1 exit temperature (isentropic):[T_2 = 873 times (1/1.69)^{0.23} = 873 times 0.868 = 758 K]After efficiency correction:[T_{2,actual} = 873 • 0.85 times (873 • 758) = 873 • 0.85 times 115 = 775 K]Stage 2 expansion (from 775 K, PR = 1.69):[T_3 = 775 times (1/1.69)^{0.23} = 775 times 0.868 = 673 K]Efficiency corrected:[T_{3,actual} = 775 • 0.85 times (775 • 673) = 775 • 0.85 times 102 = 689 K]Final outlet temperature: ~416°CTotal temperature drop: 873 − 689 = 184 KObservation:The total drop increases slightly because efficiency per stage is preserved.Each stage experiences less stress, allowing higher rotational speeds and compact design.Recuperation improves, as the exhaust gas is better matched to the compressor inlet temperature.Three-Stage Expansion[PR_{stage} = (2.86)^{1/3} = 1.41]Each stage expands more gently.Stage 1:[T_2 = 873 times (1/1.41)^{0.23} = 873 times 0.897 = 783 K]After efficiency:[T_{2,actual} = 873 • 0.85 times (873 • 783) = 873 • 76.5 = 796.5 K]Stage 2:[T_3 = 796.5 times (1/1.41)^{0.23} = 796.5 times 0.897 = 714 K]After efficiency:[T_{3,actual} = 796.5 • 0.85 times (796.5 • 714) = 796.5 • 70.2 = 726.3 K]Stage 3:[T_4 = 726.3 times 0.897 = 651 K]After efficiency:[T_{4,actual} = 726.3 • 0.85 times (726.3 • 651) = 726.3 • 63.9 = 662.4 K]Final outlet temperature: 662 K (389°C)Total drop: 873 − 662 = 211 KObservation:Multi-stage expansion produces a larger total enthalpy drop because the system approaches isentropic behavior more closely.Four-Stage Expansion[PR_{stage} = (2.86)^{1/4} = 1.30]Stage 1: 873 × (1/1.3)^{0.23} = 873 × 0.917 = 801 KActual: 873 − 0.85 × (72) = 811 KStage 2: 811 × 0.917 = 744 K → 755 KStage 3: 755 × 0.917 = 693 K → 703 KStage 4: 703 × 0.917 = 645 K → 654 KFinal outlet temperature: 654 K (381°C)Total temperature drop: ~219 KObservation:Beyond three stages, gains diminish because interstage losses and mechanical complexity offset efficiency improvements.Recuperation EffectsRecuperation recovers the turbine exhaust heat (around 380–430°C) to preheat CO2 before the main heater.With a single-stage turbine, the exhaust temperature may be too high for optimal heat recovery.Two• and three-stage turbines produce exhaust closer to the compressor discharge temperature, improving heat exchange efficiency.Four-stage systems yield slightly better thermal matching but at increased cost and complexity.Typical recuperator effectiveness ranges:Single-stage: ~75–80%Two-stage: ~85%Three• or four-stage: 90%+ (limited by pinch point and pressure drop)Summary Table| Stages | Pressure Ratio per Stage | Exit Temperature (K) | Exit Temp (°C) | ΔT Drop (K) | Typical Recuperation Effectiveness || --• | --• | -• | -• | -• | • || 1 | 2.86 | 704 | 431 | 169 | 75–80% || 2 | 1.69 | 689 | 416 | 184 | 85% || 3 | 1.41 | 662 | 389 | 211 | 88–90% || 4 | 1.30 | 654 | 381 | 219 | 90%+ |ConclusionIn a supercritical CO2 turbine:Each additional stage improves isentropic efficiency and increases total temperature drop.Three stages offer the best tradeoff between efficiency, size, and mechanical simplicity.Recuperation becomes more effective with lower exhaust temperatures, making multi-stage designs highly desirable for integrated systems.Ultimately, while pressure drives expansion, temperature (enthalpy) governs the available energy for work. Multi-stage turbines with recuperation maximize both, achieving higher efficiency and smoother thermal integration within the sCO2 Brayton cycle. |
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