Advanced propulsion systems 3 lectures

1
Advanced propulsion systems (3 lectures)

• Hypersonic propulsion background (Lecture 1)
• Why hypersonic propulsion?
• Whats different at hypersonic conditions?
• Real gas effects (non-constant CP, dissociation)
• Aircraft range
• How to compute thrust?
• Idealized compressible flow (Lecture 2)
• Isentropic, shock, friction (Fanno)
• Heat addition at constant area (Rayleigh), T, P
• Hypersonic propulsion applications (Lecture 3)
• Ramjet/scramjets
• Pulse detonation engines

2
1D steady flow of ideal gases

• Assumptions
• Ideal gas, steady, quasi-1D
• Constant CP, Cv, ? ? CP/Cv
• Unless otherwise noted adiabatic, reversible,
  constant area
• Note since 2nd Law states dS ?Q/T ( for
  reversible, gt for irreversible), reversible
  adiabatic ? isentropic (dS 0)
• Governing equations
• Equations of state
• Isentropic (S2 S1) (where applicable)
• Mass conservation
• Momentum conservation, constant area duct
• Cf friction coefficient C circumference of
  duct
• No friction
• Energy conservation
• q heat input per unit mass fQR if due to
  combustion
• w work output per unit mass

3
1D steady flow of ideal gases

• Types of analyses everything constant except
• Area (isentropic nozzle flow)
• Entropy (shock)
• Momentum (Fanno flow) (constant area with
  friction)
• Diabatic (q ? 0) - several possible assumptions
• Constant area (Rayleigh flow) (useful if limited
  by space)
• Constant T (useful if limited by materials)
  (sounds weird, heat addition at constant T)
• Constant P (useful if limited by structure)
• Constant M (covered in some texts but really
  contrived, lets skip it)
• Products of analyses
• Stagnation temperature
• Stagnation pressure
• Mach number u/c u/(?RT)1/2 (c sound speed
  at local conditions in the flow (NOT at ambient
  condition!))
• From this, can get exit velocity ue, exit
  pressure Pe and thus thrust

4
Isentropic nozzle flow

• Reversible, adiabatic ? S constant, A ?
  constant, w 0
• Momentum equation not used - simple constant-area
  form (slide 15) doesnt apply
• Recall stagnation temperature Tt temperature of
  gas stream when decelerated adiabatically to M
  0
• Thus energy equation becomes simply T1t T2t,
  which simply says that the sum of thermal energy
  (the 1 term) and kinetic energy (the (?-1)M2/2
  term) is a constant

5
Isentropic nozzle flow

• Pressure is related to temperature through
  isentropic compression law
• Recall stagnation pressure Pt pressure of gas
  stream when decelerated adiabatically and
  reversibly to M 0
• Thus the pressure / Mach number relation is
  simply P1t P2t

6
Isentropic nozzle flow

• Relation of P T to duct area A determined
  through mass conservation
• But for adiabatic reversible flow T1t T2t and
  P1t P2t also define throat area A area at M
  1 then
• A/A shows a minimum at M 1, thus it is indeed
  a throat

7
Isentropic nozzle flow

• Mass flow and velocity can be determined
  similarly

8
Isentropic nozzle flow

• Summary
• A area at M 1
• Implications
• P and T decrease monotonically as M increases
• Area is minimum at M 1 - need a throat to
  transition from M lt 1 to M gt 1 or vice versa
• mdot/A is maximum at M 1 - flow is choked at
  throat - if flow is choked then any change in
  downstream conditions cannot affect mdot
• Note for supersonic flow, M (and u) INCREASE as
  area increases - this is exactly opposite
  subsonic flow as well as intuition (e.g. garden
  hose - velocity increases as area decreases)

9
Isentropic nozzle flow
A/A
T/Tt
P/Pt
10
Isentropic nozzle flow

• When can choking occur? If M 1 or
• so need pressure ratio gt 1.89 for choking (if
  all assumptions satisfied)
• Where did Pt come from? Mechanical compressor
  (turbojet) or vehicle speed (high flight Mach
  number M1)
• Where did Tt come from? Combustion! (Even if at
  high M thus high Tt, no thrust unless Tt
  increased!) (Otherwise just reversible
  compression expansion)

11
Stagnation temperature and pressure

• From previous page
• No thrust if P1t P9t,P9 P1 T1t T9t to
  get thrust we need either
• T9t T1t, P9t P1t P1 P9 lt P1
• (e.g. tank of high-P, ambient-T gas,
• reversible adiabatic expansion)
• B. T9t gt T1t, P9t P1t gt P1 P9
• (e.g. high-M ramjet/scramjet,
• no Pt losses)
• C. T9t gt T1t, P9t gt P1t P1 P9
• (e.g. low-M turbojet or fan)
• Fan T9t/T1t (P9t/P1t)(?-1)/?
• due to adiabatic compression
• Turbojet T9t/T1t gt (P9t/P1t)(?-1)/?
• due to adiabatic compression
• plus heat addition
• Could get thrust even with

P9t P1t P9 lt P1 T9t T1t
P1 P1t (M1 0)
Case A
P9t P1t P9 P1 T9t gt T1t
P1t gt P1 (M1 gt 0)
Case B
P9t gt P1t P9 P1 T9t gt T1t
P1t P1 (M1 0)
Case C
12
Stagnation temperature and pressure

• Note also ?(Tt) heat or work transfer

13
Constant everything except S (shock)

• Q what if A constant but S ? constant? Can
  anything happen while still satisfying mass,
  momentum, energy, eqn. of state?
• A YES! (shock)
• Implications
• One possibility is no change in state ( )1 ( )2
• If M1 gt 1 then M2 lt 1 and vice versa - equations
  dont show a preferred direction, but only M1 gt
  1, M2 lt 1 results in dS gt 0, thus M1 lt 1, M2 gt 1
  is impossible
• Tt constant (no change in total enthalpy) but Pt
  decreases across shock (a lot if M gtgt 1!), P, T
  increase
• Note there are only 2 states, ( )1 and ( )2 - no
  continuum of states

14
Constant everything except S (shock)
T2/T1
P2/P1
T2t/T1t
M2
P2t/P1t
15
Everything constant except momentum (Fanno flow)

• Since friction loss is path dependent, need to
  use differential form of momentum equation
  (constant A by assumption)
• Combine and integrate with differential forms of
  mass, energy, eqn. of state from Mach M to
  reference state ( ) at M 1 (not a throat in
  this case since constant area!)
• Implications
• Stagnation pressure always decreases towards M
  1
• Cant cross M 1 with constant area with
  friction!
• M 1 corresponds to the maximum length (L) of
  duct that can transmit the flow for the given
  inlet conditions (Pt, Tt) and duct properties
  (C/A, Cf)

16
Everything const. but momentum (Fanno flow)

• What if neither the initial state (1) nor final
  state (2) is the choked () state? Again use
  P2/P1 (P2/P)/(P1/P) etc., except for L, where
  we subtract to get net length ?L

17
Everything constant except momentum
Pt/Pt
Tt/Tt
T/T
Length
P/P
Length
18
Everything constant except momentum
Pt/Pt
P/P
Length
Tt/Tt
T/T
Length
19
Heat addition at constant area (Rayleigh flow)

• Mass, momentum, energy, equation of state all
  apply
• Reference state ( ) use M 1 (not a throat in
  this case!)
• Energy equation not useful except to calculate
  heat input (q Cp(T2t - T1t)) or dimensionless
  q/CPTt 1 - Tt/Tt)
• Implications
• Stagnation pressure always decreases towards M
  1
• Stagnation temperature always increases towards M
  1
• Cant cross M 1 with constant area heat
  addition!
• M 1 corresponds to the maximum possible heat
  addition
• but theres no particular reason we have to keep
  area (A) constant when we add heat!

20
Heat addition at const. Area (Rayleigh flow)

• What if neither the initial state (1) nor final
  state (2) is the choked () state? Again use
  P2/P1 (P2/P)/(P1/P) etc.

21
Heat addition at constant area
Pt/Pt
Tt/Tt
T/T
P/P
22
T-s diagram - reference state M 1
Fanno
M lt 1
Shock
Rayleigh
M lt 1
M gt 1
23
T-s diagram - Fanno, Rayleigh, shock
Rayleigh
Shock
Constant area, with friction, no heat addition
M lt 1
Constant area, no friction, with heat addition
Fanno
M lt 1
M gt 1
This jump constant area, no friction, no heat
addition ? SHOCK!
M gt 1
24
Heat addition at constant pressure

• Relevant for hypersonic propulsion if maximum
  allowable pressure (i.e. structural limitation)
  is the reason we cant decelerate the ambient air
  to M 0)
• Momentum equation AdP mdotdu 0 ? u
  constant
• Reference state ( ) use M 1 again but nothing
  special happens there
• Again energy equation not useful except to
  calculate q
• Implications
• Stagnation temperature increases as M decreases,
  i.e. heat addition corresponds to decreasing M
• Stagnation pressure decreases as M decreases,
  i.e. heat addition decreases stagnation T
• Area increases as M decreases, i.e. as heat is
  added

25
Heat addition at constant pressure

• What if neither the initial state (1) nor final
  state (2) is the reference () state? Again use
  P2/P1 (P2/P)/(P1/P) etc.

26
Heat addition at constant P
Pt/Pt
P/P
Tt/Tt
T/T, A/A
27
Heat addition at constant temperature

• Probably most appropriate case for hypersonic
  propulsion since temperature (materials) limits
  is usually the reason we cant decelerate the
  ambient air to M 0
• T constant ? c (sound speed) constant
• Momentum AdP mdotdu 0 ? dP/P ?MdM 0
• Reference state ( ) use M 1 again
• Implications
• Stagnation temperature increases as M increases
• Stagnation pressure decreases as M increases,
  i.e. heat addition decreases stagnation T
• Minimum area (i.e. throat) at M ?-1/2
• Large area ratios needed due to exp term

28
Heat addition at constant temperature

• What if neither the initial state (1) nor final
  state (2) is the reference () state? Again use
  P2/P1 (P2/P)/(P1/P) etc.

29
Heat addition at constant T
Tt/Tt
A/A
T/T
Pt/Pt
P/P
30
T-s diagram for diabatic flows
Const P
Const T
Rayleigh (Const A)
Rayleigh (Const A)
31
Pt vs. Tt for diabatic flows
Rayleigh (Const A)
Const P
Rayleigh (Const A)
Const T
32
Area ratios for diabatic flows
Const T
Const T
Const P
Const A
33
Summary

• Choking - mass, heat addition at constant area,
  friction with constant area - at M 1
• Supersonic results usually counter-intuitive
• If no friction, no heat addition, no area change
  - its a shock!
• Which is best way to add heat?
• If maximum T or P is limitation, obviously use
  that case
• What case gives least Pt loss for given increase
  in Tt?
• Minimize d(Pt)/d(Tt) subject to mass, momentum,
  energy conservation, eqn. of state
• Result (Lots of algebra - many trees died to
  bring you this result)
• Adding heat (increasing Tt) always decreases Pt
• Least decrease in Pt occurs at lowest possible M

34
Summary
35
Summary of heat addition processes