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Turbines

Turbines. Thermodynamics Professor Lee Carkner Lecture 18. Ideal gas law P v = RT P v /T = P v /T Isentropic (constant specific heats) (T 2 /T 1 ) = ( v 1 / v 2 ) k-1 (T 2 /T 1 ) = (P 2 /P 1 ) (k-1)/k (P 2 /P 1 ) = ( v 1 / v 2 ) k q = 0 Isentropic (variable specific heats)

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Turbines

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  1. Turbines Thermodynamics Professor Lee Carkner Lecture 18

  2. Ideal gas law Pv = RT Pv/T = Pv/T Isentropic (constant specific heats) (T2/T1) = (v1/v2)k-1 (T2/T1) = (P2/P1)(k-1)/k (P2/P1) = (v1/v2)k q = 0 Isentropic (variable specific heats) (P2/P1) = (Pr2/Pr1) (v2/v1) = (vr2/vr1) Pr and vr are functions of T and you can use to look up u and h Isochoric v = v q = cvDT = Du Isobaric P = P q = cpDT = Dh w = PDv Isothermal T = T q = w Engine Cheat Sheet

  3. PAL # 17 Gas Power Cycles • Otto engine: rc = 8, qin = 750 kJ/kg, P1 = 95 kPa and T1 = 300 K, R = 0.287, k =1.4, cv = 0.718 • Pv = RT, v1 = RT1/P1 = (0.287)(300)/(95) = 0.906 m3/kg = vmax = v4 • rc = vmax/vmin, vmin = vmax/rc = 0.906/8 = 0.113 = v2 = v3 • T2 = T1(v1/v2)k-1 = (300)(8)0.4 = 689 K

  4. PAL # 17 Gas Power Cycles • DT = q/cv, T3 = (q/cv)+T2 = (750/0.718)+689 =1734 • T4 = T3(v3/v4)k-1 = (1734)(1/8)0.4 = 755 • P3 = (0.287)(1734)/ 0.113 = 4404 kPa • Find heats and work • qout = cv(T4-T1) = (0.718)(755-300) = 327 kJ/kg • Find efficiency and MEP • hth = w/qin = 423/750 = 0.564 = 56.4% • MEP = w/(vmax-vmin) = 423/(0.906-0.113) = 533 kPa

  5. Brayton Cycle • Used for electricity generation and propulsion • Ideal, air standard, Brayton cycle: • Isentropic compression (compressor) • Isentropic expansion (turbine) • We replace ignition and exhaust with heat input and output to make a closed cycle

  6. Brayton Efficiency • The heat in and out take place during constant pressure processes so: • We can use this to write the efficiency as: hth,Brayton = 1 – qout/qin = 1 – (T4-T1)/(T3-T2) • We can relate the temperatures to the pressures with the isentropic relation: • (T2/T1) = (P2/P1)(k-1)/k • And thus the efficiency is hth,Brayton = 1 – 1/rp(k-1)/k

  7. Brayton Turbines • Higher pressure ratio produces more efficiency • Limited by the temperature the turbine blades can withstand • Generally have better power to weight ratios than reciprocating engines, but are more expensive and use more fuel

  8. Regeneration • We can use them to heat the input gases • Such a system called a regenerator • Extra efficiency not always worth the expense of the installation and operation costs

  9. Effectiveness • The gas from the compressor enters the regenerator at T2 and leaves at T5 • We can compare the actual energy gain with the best case with the effectiveness, e • e = • For cold air standard, Dh = cpDT so • e = (T5-T2)/(T4-T2)

  10. Regeneration Efficiency • Effectiveness generally around 0.85 • The efficiency of a turbine with regeneration is: hth,regen = 1 – (T1/T3)rp(k-1)/k • Regeneration most effective at low rp and when the difference between the temperatures is high

  11. Multistage • We do this by using multistage processes where the fluid is heated or cooled between stages • Called intercooling or reheating • Depending on the number of stages • More stages makes the device for complicated and expensive

  12. Multistage Properties • The ratio of the input and output pressures of the compressors are equal P2/P1 = P4/P3 P6/P7 = P8/P9 • The input temperatures and enthalpies are the same of each compressor T1 = T3, h1 = h3

  13. Jet Propulsion • If we reduce the work produced by a turbine to just enough to power the compressor, the exit gases will have a lot of energy • Used for aircraft propulsion • Amount of force (or thrust) is just the difference between the input and output momentum

  14. Jet Efficiency • If the wind velocity is small, Vinlet is equal to the aircraft’s speed • The propulsive power of a turbojet is just: Wp’ = FVaircraft = m’(Vexit-Vinlet)Vaircraft hp = W’p/Q’in

  15. Turbofan Has a turbine powered fan up front to move more air through Afterburner Ignite exhaust gases for burst of speed Ramjet A engine that is all afterburner Scramjet Supersonic ramjet Rocket A ramjet with its own air supply Kinds of Jets

  16. Next Time • Read: 10.1-10.5 • Homework: Ch 9, P: 95, 121, Ch 10, P: 15, 34

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