AME 436 Energy and Propulsion

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AME 436Energy and Propulsion

• Lecture 12
• Propulsion 3 Ideal performance of turbojets

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Outline

• Turbojet analysis - assumptions and goals
• Process summary
• State-by-state analysis
• Results
• Thrust
• Efficiency
• Fuel consumption
• Effects of
• Compressor pressure ratio
• Flight Mach number
• ?? limit

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Turbojet analysis

• Assumptions
• Steady, quasi-1D
• Constant CP, ?, Pexit Pambient (P9 P1 in
  current notation)
• Isentropic except for heat addition process
• Heat addition at M ltlt 1, FAR ltlt 1 up to materials
  limit temperature T?
• Goals determine (defined later) Specific
  Thrust, Thrust Specific Fuel Consumption and
  thermal efficiency as a function of flight Mach
  number M1, turbine inlet temperature limit T?
  compressor pressure ratio P3t/P2t

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Why use Brayton cycle to model gas turbines?

• Pressure compression ratio (r) pressure
  expansion ratio (Pe Pa), which corresponds to
  best possible thrust (Lecture 10)
• Heat input at constant pressure realistic for
  steady-flow, M ltlt 1 process (see Rayleigh flow
  analysis)
• As always, constant s compression/expansion
  corresponds to an adiabatic and reversible
  process - not true but not bad either
• Notes on Brayton cycle P-v and T-s diagrams
• v on P-v diagram is specific volume (v) (m3/kg)
  which IS a property of the gas (we cant use
  cylinder volume V as in unsteady-flow engines
  since not a fixed mass of material in a changing
  cylinder volume)
• s is specific entropy (J/kg-K) which IS a
  property of the gas, heat transfer ?Tds if mass
  doesnt change during heat addition
• P-v diagrams not as useful as with unsteady-flow
  engines where we can use a cylinder pressure
  gauge to measure P vs. t and calculate V vs. t
  from crank angle to get P-V diagram for
  comparison with ideal cycle (would need a
  pressure gauge moving along with the flow!)

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Ideal turbojet cycle - process summary
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P-V T-s diagrams for ideal turbojet

• Model shown is open cycle, where mixture is
  inhaled, compressed, burned, expanded then thrown
  away (not recycled)
• In a closed cycle with a fixed (trapped) mass of
  gas to which heat is transferred to/from, 9 ? 1
  would be connected (Why dont we do this? Heat
  transfer is too slow!)

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P-V T-s diagrams for ideal turbojet
Turbine work Cp(T3-T2)
Heat input Cp(T4-T3)
P constant
KE out Cp(T5-T9)
Compressor work Cp(T3-T2)
P constant
KE in Cp(T2-T1)
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Ideal turbojet cycle - analysis

• Inlet conditions M1, T1, P1 after diffuser (2)
  decelerate to M2 0
• After compressor (3) isentropic compression by
  pressure ratio ?c
• After combustor (4) constant-pressure heat
  addition to T?
• After turbine (5) isentropic expansion to pay
  for compressor work

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Ideal turbojet cycle - analysis

• After nozzle (9) isentropic expansion to P9 P1
  (Pe Pa)

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Ideal turbojet cycle - Thrust

• P9 P1 (Pe Pa), FAR ltlt 1 Specific Thrust (ST)
  ? Thrust/mdotac1
• This gives the thrust in terms of
• Air flow (mdota)
• Sound speed at ambient conditions (c1)
• Flight Mach number M1 and ?r ? 1 (?-1)/2M12
• Compressor pressure ratio ?c
• Materials limited temperature at turbine inlet
  ??T1
• And we assumed ideal gas, constant specific
  heats, FAR ltlt 1, Pe Pa, isentropic compression
  and expansion, constant-P combustion at M 0,
  add heat to ?? materials limit

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Ideal turbojet cycle - notes on thrust

• ?c 1 (ramjet, no compressor)
• If ?? ?r(?c)(?-1)/ ? then Thrust 0 ??
  materials limited temperature reached just by
  decelerating the gas to M 0 and compressing it
  in the compressor so no head room in terms of
  temperature to enable heat addition) this could
  happen for very (unrealistically) high ?c, or at
  very high M1 (thus high ?r)
• Since either too low or too high ?c leads to
  Thrust 0, there is a value of ?c that maximizes
  Thrust (but not any flavor of efficiency)
• At this condition, T3 T9 (??)1/2T1, i.e.
• temperature at end of compression
• temperature at end of expansion
• For typical ?? 5, ?r 1.128 (M 0.8),
• ? 1.4, this corresponds to ?c 10.97

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Ideal turbojet cycle - notes on thrust

• What if Pe ? Pa or FAR is not ltlt 1?

Valid for any 1D steady propulsion system if
working fluid is an ideal gas with constant CP, ?
New term for Pe ? Pa
New term for FAR not ltlt 1
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Ideal turbojet cycle - thermal efficiency
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Ideal turbojet cycle - thermal efficiency

• Note that thermal efficiency ?th 1 -
  1/r(?-1)/?, where r ?r?c, i.e. the combined
  pressure rise due to ram effect compression
  (decelerating the gas from flight mach number M1
  to M 0) AND the mechanical compression - each
  has the same effect on thermal efficiency
• This result is very similar to the Otto cycle
  (?th 1 - 1/r(?-1), where r is the volume (not
  pressure) ratio) why the difference? Otto is
  constant volume heat addition and expansion back
  to the initial volume, whereas, Brayton is
  constant pressure heat addition and expansion
  back to the initial pressure
• In either case ?th 1 - TL/TH and TL/TH is the
  same for each Carnot strip in this cycle TL/TH
  (PL/PH)(?-1)/? (VH/VL)(?-1), thus ?th 1 -
  (PL/PH)(?-1)/? 1 - (VH/VL)(?-1) the only
  difference is that Otto cycles are specified in
  terms of volume ratio whereas Brayton cycles are
  specified in terms of pressure

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Ideal turbojet cycle - fuel consumption

• Thrust Specific Fuel Consumption (TSFC) (PDRs
  definition)
• (Usual definition of TSFC is just
  mdotfuel/Thrust, but this is not dimensionless
  use QR to convert mdotfuel to heat input, one can
  use either u1 or c1 to convert the denominator to
  a quantity with units of power, but using u1 will
  make TSFC blow up at u1 0, i.e. at takeoff)
• The term (Thrust/mdotac1) is the specific thrust,
  already computed all we need to do to get TSFC
  is to compute FAR energy balance on combustor
  (heat input change in total enthalpy)

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Ideal turbojet cycle - fuel consumption

• Note on FAR how do we know that we can add
  enough fuel to reach the ?? limit before we run
  out of O2 in the air? We know from the energy
  balance on the combustor,
• Using realistic numbers FARstoich 0.068, QR
  4.5 x 107 J/kg, CP 1400 J/kgK, T1 300K,
  (FARstoichQR)/(CPT1) 7.3, and we require 0 lt
  FAR lt FARstoich, thus at stoichiometric,
• But typically the maximum allowable turbine
  inlet temperature T4t is 1500K, so with T1
  300K, ?? 5 lt 7.3, so we can never add the
  stoichiometric amount of fuel - we reach the
  materials limit first

Ideal turbojet use specific thrust
Thrust/mdotac1 from page 10
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Ideal turbojet cycle - fuel consumption

• Also note that FAR can be calculated via
• For ?c 30, ? 1.4, ?? 4.3, ?r 1 (M1 0),
  QR 4.5 x 107 J/kg, CP 1400 J/kgK, T1 300K,
  FAR 0.0155 ? FAR/FARstoich 0.23 ltlt 1
• Wait - isnt this too lean to burn? For premixed
  flame, yes, for non-premixed flame (e.g. spray
  flame, like diesel but continuous, not at
  discrete times), not a problem
• Also recall ?overall M1/TSFC
• Another measure of fuel consumption Specific
  Impulse (ISP) (see lecture 10) thrust per unit
  weight flow rate of fuel, units of seconds)

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Ideal turbojet cycle - results - effect of ?c

• See baseline conditions on Master sheet within
  worksheet key values M1 0.8, ? 1.4, ?? 5,
  ?c 30 (one parameter changed, others fixed on
  each of the sheets labeled Mach, Pi_c, etc.)
• For very low compressor pressure ratio ?c,
  thermal efficiency is low, so both thrust and
  TSFC are low
• At very (unrealistically) high ?c, very little
  fuel can be added, thus Thrust decreases, but
  TSFC is great!

M1 0.8 ? 1.4 ?? 5 ?c varies
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Ideal turbojet cycle - results - effect of ?c

• T-s diagrams show tall skinny T-s diagrams for
  high ?c, banana shaped cycles for low ?c, and
  fat cycles for intermediate ?c

?? limit
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Ideal turbojet cycle - results - effect of M1

• For high M1, specific thrust decreases since less
  fuel can be added (?? limit again)
• ?th increases as M1 increases since total
  pressure ratio ?r?c increases (?r ?r?/(?-1),
  ?r 1 (?-1)/2M12)
• Also ?prop 2(u1/u9)/(1 u1/u9) increases as M1
  increases since u1/u9 ? 1
• TSFC increases even though ?overall ?th?prop
  increases since PDRs definition of TSFC has M1
  in it - biases results against high M1

M1 varies ? 1.4 ?? 5 ?c 30
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Ideal turbojet cycle - results - effect of M1

• T-s diagrams similar to ?c effect - tall skinny
  diagrams for high M1, banana shaped cycles for
  low M1, and fat cycles for intermediate M1

?? limit
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Ideal turbojet cycle - results - effect of ??

• As ?? increases, specific thrust increases but so
  does TSFC due to lower propulsive efficiency
  (?prop 2(u1/u9)/(1 u1/u9) for fixed u1, u9
  increases with increasing ??)
• At very low ??, no heat addition is possible,
  thus no thrust

M1 0.8 ? 1.4 ?? varies ?c 30
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Ideal turbojet cycle - results - effect of ??

• T-s diagrams just show increasing heat addition
  as ?? increases, no change in ?th (which can be
  seen on T-s) but decreases in ?prop (which cant
  be seen on T-s or P-v)

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Summary - ideal turbojets

• Compress isentropically, burn isobarically,
  expand isentropically back to ambient pressure
• Matching conditions compressor work turbine
  work P1 P9 (i.e. Pexit Pambient)
• Turbojet performance is limited by compressor
  pressure ratio ?c and turbine inlet temperature
  limit (??)
• Low ?c thermal efficiency low, thrust low
• High ?c thermal efficiency high but not much
  fuel can be added before ?? limit reached so
  again thrust low (also propulsive efficiency low
  because of high exit M)
• Intermediate ?c thermal efficiency intermediate,
  thrust highest
• Increasing ?? always increases thrust but
  propulsive efficiency suffers