AME 436 Energy and Propulsion

1
AME 436Energy and Propulsion

• Lecture 6
• Unsteady-flow (reciprocating) engines 2
• Using P-v and T-s diagrams

2
Outline

• Air cycles
• What are they?
• Why use P-v and T-s diagrams?
• Using P-V and T-s diagrams for air cycles
• Seeing heat, work and KE
• Constant P and v processes
• Inferring efficiencies
• Compression expansion component efficiencies
• Correspondence between processes on P-v and T-s
• Hints tricks

3
Air-cycles - what are they?

• In this course we will work primarily with air
  cycles in which the working fluid is treated as
  just air (or some other ideal gas)
• In air cycles, changes in gas properties (CP, Mi,
  ?, etc.) due to changes in composition,
  temperature, etc. are neglected this greatly
  simplifies the analysis and leads to simple
  analytic expressions for efficiency, power, etc.
• Later well examine fuel-air cycles (using
  GASEQ) where the real gases are considered and
  the properties change as composition,
  temperature, etc. change (but still cant account
  for slow burn, heat loss, etc. since its still a
  thermodynamic analysis that tells us nothing
  about reaction rates, burning velocities, etc.)
• In addition to the analytical results, P-V and
  T-s diagrams will be used extensively to provide
  a visual representation of cycles

4
Why use P-v diagrams?

• Pressure vs. time and cylinder volume vs. time
  are easily measured in reciprocating-piston
  engines
• ? PdV work
• ? PdV over whole cycle net work transfer net
  change in KE net change in PE net heat
  transfer
• Heat addition is usually modeled as constant
  pressure or constant volume, so show as straight
  lines on P-V diagram

5
Why use T-s diagrams?

• Idealized compression expansion processes are
  constant S since dS ?Q/T for adiabatic process
  ?Q 0, for reversible (not gt) sign applies,
  thus dS 0 (note that dS 0 still allows for
  any amount of work transfer to occur in or out of
  the system, which is what compression expansion
  processes are for)
• For reversible process, ?TdS Q, thus area
  under T-s curves show amount of heat transferred
• ? TdS over whole cycle net heat transfer net
  work transfer net change in KE net change in
  PE
• T-s diagrams show the consequences of non-ideal
  compression or expansion (dS gt 0)
• For ideal gases, ?T heat xfer or work xfer or
  ?KE (see next 2 slides)
• Efficiency can be determined by breaking any
  cycle into Carnot-cycle strips, each strip (i)
  having ?th,i 1 - TL,i/TH,i

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T-s P-v for control mass work, heat KE

• For an ideal gas with constant CP Cv
• ?h Cp?T, ?u Cv?T
• and the 1st Law says, for a control mass with ?PE
  0 (in internal combustion engines we can almost
  always neglect PE)
• dE d(U KE) dm(u V2/2) ?q - ?w
• q Q/m (heat transfer per unit mass)
• w W/m (work transfer per unit mass)
• If no work transfer (dw 0) or KE change (dKE
  0), du CvdT dq
• ? q1?2 Cv(T2 - T1)
• If no heat transfer (dq 0) or ?KE, du CvdT
  dw
• ? w1?2 -Cv(T2 - T1)
• If no work or heat transfer
• ? ?KE V22/2 - V12/2 -Cv(T2 - T1)
• For a control mass containing an ideal gas with
  constant Cv,
• ?T heat transfer - work transfer - ?KE
• Note that the 2nd law was not invoked, thus the
  above statements are true for any process,
  reversible or irreversible

7
T-s P-v for control volume work, heat KE

• 1st Law says, for a control volume, steady flow,
  with ?PE 0
• For an ideal gas with constant CP, dh CPdT ? h2
  - h1 CP(T2 - T1)
• If 2 outlet, 1 inlet, and noting
• If no work transfer (dw 0) or KE change (dKE
  0), du CvdT dq
• ? q1?2 CP(T2 - T1)
• If no heat transfer (dq 0) or ?KE, du CvdT
  dw
• ? w1?2 -CP(T2 - T1)
• If no work or heat transfer
• ? ?KE V22/2 - V12/2 -CP(T2 - T1)
• For a control volume containing an ideal gas with
  constant CP,
• ?T heat transfer - work transfer - ?KE
• (Same statement as control mass with CP
  replacing Cv)
• Again true for any process, reversible or
  irreversible

8
T-s P-v diagrams work, heat KE
3
w3?4 (V42/2 - V32/2) -Cv(T4-T3) (control
mass) w3?4 (V42/2 - V32/2) -CP(T4-T3)
(control vol.)
q3?2 Cv(T3-T2) (const. V) q3?2 CP(T3-T2)
(const. P)
4
2
w1?2 (V22/2 - V12/2) -Cv(T2-T1) (control
mass) w1?2 (V22/2 - V12/2) -CP(T2-T1)
(control vol.)
1
Case shown cons. vol. heat in, rc re 3, ?
1.4, ?Tcomb fQR/Cv 628K, P1 0.5 atm
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T-s P-v diagrams work, heat, KE PE

• Going back to the 1st Law again
• Around a closed path, since E U KE PE
  since U is a property of the system,
  , thus around a closed path, i.e. a complete
  thermodynamic cycle (neglecting PE again)
• But wait - does this mean that the thermal
  efficiency
• No, the definition of thermal efficiency is
• For a reversible process, ?Q TdS, thus ?q Tds
  and
• Thus, for a reversible process, the area inside a
  cycle on a T-s diagram is equal to (net work
  transfer net KE) and the net heat transfer

10
T-s P-v diagrams work, heat KE

• Animation using T-s diagram to determine heat
  work

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Constant P and V curves

• Recall for ideal gas with constant specific heats
  (1st lecture)
• If P constant, ln(P2/P1) 0 ? T2
  T1exp(S2-S1)/CP
• If v constant, ln(v2/v1) 0 ? T2
  T1exp(S2-S1)/Cv
• ? constant P or v curves are growing
  exponentials on a T-s diagram
• Since constant P or v curves are exponentials, as
  s increases, the ?T between two constant-P or
  constant-v curves increases as shown later, this
  ensures that compression work is less than
  expansion work for ideal Otto or Brayton cycles
• Since CP Cv R, CP gt Cv or 1/CP lt 1/Cv,
  constant v curves rise faster than constant P
  curves on a T-s diagram
• Constant P or constant v lines cannot cross

12
Constant P and V curves
3
T(s) T2exp(s-s2)/Cv (const vol.) T(s)
T2exp(s-s2)/CP (const press.)
4
2
Constant v or P curves spread out as s increases
? T3 - T4 gt T2 - T1
1
T(s) T1exp(s-s1)/Cv (const vol.) T(s)
T1exp(s-s1)/CP (const press.)
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Constant P and V curves
3
Payback for compression work and KE decrease
(T3 - T4) - (T2 - T1) net work KE increase
4
2
T2 - T1 Work input KE decrease during
compression
1
Net work KE decrease (T3 - T4) - (T2 - T1)
(T2 - T1)exp(?s/Cv) - 1 gt 0 or (T2 -
T1)exp(?s/CP) - 1 gt 0
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Constant P and V curves
Double-click chart to open Excel spreadsheet
Constant-v curves are steeper than constant-P
curves on the T-s Both cases T2/T1 1.552, ?
1.4, fQR 4.5 x 105 J/kg, P1 0.5 atm The two
cycles shown also have the same thermal
efficiency ?th
15
Inferring efficiencies
TH,i
Carnot cycle strip ?th,i 1 - TL,i/TH,i
TL,i
Double-click chart to open Excel spreadsheet
Carnot cycles appear as rectangles on the T-s
diagram any cycle can be broken into a large
number of tall skinny Carnot cycle strips, each
strip (i) having ?th,i 1 - TL,i/TH,i
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Compression expansion efficiency

• If irreversible compression or expansion, dS gt
  ?Q/T if still adiabatic (?Q 0) then dS gt 0
• Causes more work input (more ?T) during
  compression, less work output (less ?T) during
  expansion
• Define compression efficiency ?comp expansion
  efficiency ?exp

17
Compression expansion efficiency

• Control volume replace Cv with CP, but it
  cancels out so definitions are same
• These relations give us a means to quantify the
  efficiency of an engine component (e.g.
  compressor, turbine, ) or process (compression,
  expansion) as opposed to the whole cycle

18
Compression expansion efficiency

• Animation comparison of ideal Otto cycle with
  non-ideal compression expansion
• Same parameters as before but with ?comp ?exp
  0.9

2 charts on top of each other double-click each
to open Excel spreadsheets
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Correspondence between P-v T-s
20
Correspondence between P-v T-s
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Using T-s and P-v diagrams - summary

• Thermodynamic cycles as they occur in IC engines
  are often approximated as a series of processes
  occurring in an ideal gas
• T-s and P-v diagrams are very useful for
  inferring how changes in a cycle affect
  efficiency, power, peak P T, etc.
• The ?T (on T-s diagrams) and areas (both T-s
  P-v) are very useful for inferring heat work
  transfers
• Each process (curve or straight line) on T-s or
  P-v diagram has of 3 parts
• An initial state
• A process (const. P, v, T, s, as shown in the
  previous slides), constant area (Rayleigh, Fanno
  or shock flow, discussed in propulsion section),
  etc.
• A final state, which is usually
• For compression and expansion processes in
  reciprocating piston engines, a specified volume
  ratio relative to the initial state
• For compressors in propulsion cycles, a specified
  pressure ratio
• For turbines in propulsion cycles, a specified
  temperature that makes the work output from the
  turbine equal the work required to drive
  compressor and/or fan
• For diffusers in propulsion cycles, a specified
  Mach number (usually zero)
• For nozzles in propulsion cycles, the pressure
  after expansion (usually the ambient pressure)
• For heat addition processes, either a specified
  heat input ? Tds (i.e. a mixture having a
  specified FAR and QR) thus a given area on the
  T-s diagram, or a specified temperature (i.e. for
  temperature limited turbines in propulsion
  cycles)
• The constant P and constant v exponential curves
  on the T-s diagram are very useful for
  determining end states

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Using T-s and P-v diagrams - summary

• Three or more processes combine to make a
  complete cycle
• When drawing P-v or T-s diagrams, ask yourself
• What is the P, v, T and s of the initial state?
  Is it different from the baseline case?
• For each subsequent process
• What is the process? Is it the same as the
  baseline cycle, or does it change from (for
  example) reversible to irreversible compression
  or expansion? Does in change from (for example)
  constant pressure heat addition to heat addition
  with pressure losses?
• When the process is over? Is the target a
  specified pressure, volume, temperature, heat
  input, work output, etc.?
• Is a new process (afterburner, extra turbine work
  for fan, etc.) being added or is existing one
  being removed?
• In gas turbine cycles, be sure to make work
  output of turbines work input to compressors
  and fans in gas
• Be sure to close the cycle by having (for
  reciprocating piston cycles) the final volume
  initial volume or (for propulsion cycles)
  (usually) the final pressure ambient pressure