1.3 Ideal Rocket Theory (part 1) - TU Delft OCW · 2016. 12. 4. · Ideal Rocket Theory assumptions...
Transcript of 1.3 Ideal Rocket Theory (part 1) - TU Delft OCW · 2016. 12. 4. · Ideal Rocket Theory assumptions...
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Ideal Rocket Theory (part 1)
1.3
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Why an “ideal” rocket theory?
• Our objective is to find simplified equations for :
• The jet velocity ve
• The mass flow rate
• The exit pressure pe
• This can be achieved with the Ideal Flow Theory , consisting of :
• A simplified rocket geometry
• A set of physical assumptions
m
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Ideal rocket geometry
(Combustion) chamber
•High pressure
• (High temperature)
•Low speed
Convergent-Divergent Nozzle
•Propellant is accelerated
•No external energy is provided
In some rockets, no
combustion takes place
→ propellant in the chamber
is at low temperature
Nozzle Inlet
(Chamber conditions)
Nozzle Throat
Nozzle Exit
Nozzle Expansion Ratio
* eA
A
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Ideal Rocket Theory assumptions
1. The propellant is a perfect gas2. The propellant is a calorically ideal gas3. Propellant has constant homogeneous chemical composition4. Nozzle flow is steady (not dependant on time)5. Nozzle flow is isentropic (no energy is provided or lost)6. Nozzle flow is 1-dimensional (quantities vary only along axis)7. Flow velocity is purely axial8. The propellant experiences no external forces in the nozzle9. Propellant in the chamber has negligible velocity (vc 0)
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Ideal Rocket Theory building blocksIdeal Gas EquationsConservation Equations
Mass
Momentum
Energy
constant
p
2
A
W
R pa T
M
A
W
Rp T
M
1
Ap
W
Rc
M ph c T
constant v A m
21constant
2 p v
21constant
2 h v
vM
a
= density [kg/m3]
v = velocity [m/s]
A = nozzle area [m2]
p = pressure [Pa]
h = enthalpy [m2/s2]
T = temperature [K]
RA = universal gas constant = 8314 J/(K*kmol)
cp = constant pressure specific heat [J/K*kg]
= specific heat ratio [-]
MW = molecular mass [kg/kmol]
a = speed of sound [m/s]
M = Mach number [-]
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Two more assumptions…
pC TC v 0
M 0MW cp
1. Propellant conditions in the chamber (TC , pC ) are known
2. Propellant composition and characteristics ( , cp , MW , constant through the nozzle) are known
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Why a convergent-divergent nozzle?
2 1 dA dv
MA v
v 0
M 0
• Convergent (dA < 0) → dv > 0 only if M < 1 (subsonic flow)
• Divergent (dA > 0) → dv > 0 only if M > 1 (supersonic flow)
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Why a convergent-divergent nozzle?
2 1 dA dv
MA v
v 0
M 0
To accelerate the flow everywhere in the nozzle (dv > 0):
• Subsonic convergent (M < 1), supersonic divergent (M > 1)
• Sonic throat (M = 1)
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Jet velocity
1
21
1
eAe C
W C
pRv T
M p
• High jet velocity is obtained
with:
High chamber temperature TC
Low molecular mass MW
Low pressure ratio pe / pc
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Mass flow rate
*
C
AC
W
p Am
RT
M
• = Vandenkerckhove function
• High mass flow rate is obtained with:
Low chamber temperature TC
High molecular mass MW
High chamber pressure pc
High throat area A*
• For given TC , pC , , MW , A*, only one
mass flow rate makes sonic throat
conditions possible (chocked flow)
1
11
2
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Exit pressure
*
2 1
21
1
e
e e
C C
A
Ap p
p p
• For given , nozzle geometry
(expansion ratio ) and
pressure ratio pe / pC are
directly related
• Implicit equation for pe / pC
as a function of
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Exit pressure
• Weak dependence on
• High → high pC / pe → lower
pe for a given pC
• For given expansion ratio and
chamber pressure, the exit
pressure pe is fixedpC / pe
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Nozzle expansion conditions
• Three cases are possible:
1) pe < pa → over-expanded nozzle
2) pe = pa → adapted nozzle
3) pe > pa → under-expanded nozzle
• In cases (1) and (3), pressure adjusts to
ambient conditions through shock
waves outside the nozzle
• For a given nozzle geometry, thrust is
maximum when nozzle is adapted