ACCELERATION & MASS LOSS:
At any instant during burn time:
 dm is ejected at ue leading to du=adt

Fx 
d
u x dV   u x dm

cv
cs
dt
dt  pe  pa  Ae  D  g cos      u  du   u  dm  u  ue   dm  u
d
dm  ue  mdt  ue  
dt  ue
dt
du
 dmue
du   pe  pa  Ae  mue  D  g cos   dt
d D
 dt  g cos  dt


 
 
 o 
u  u g  D 0  ueq ln 
 ub  ueq ln  o 

 In space:
  
 b 





Velocity reached depends only on fuel burned!
 du  ueq
SINGLE-STAGE ROCKET EXAMPLE:

For a vertical rocket the height reached during burning is:
tb
hb   udt
0

u  u D0  ub  ueq ln    g (h(t ))cos  dt
tb
0
The instantaneous velocity (negligible drag, constant ue, h<<Re,
pe=pa)
  t  
u(t ) D 0  ue ln 
  g et
 o 

At constant burn-rate:
 hb  uetb

  t   o   o   b 
ln  
1
 uetb  g etb2
1 
2
 t 
t
1t


 1  1  
tb
o
   tb
Conversion of the total kinetic energy to height (Bernoulli):
hmax
ub2
 hb 
2 ge
ue2 ln  
 


 uetb 
ln     1
2 ge
  1

2
 hmax
OPTIMAL BURNING TIME:

As we saw the maximum height is:
ue2 ln  
 


 uetb 
ln     1
2 ge
  1

2
hmax

Which shows that the shorter the burning time and
higher the acceleration in a gravity field the higher the
efficiency
ln  
1 2
hb  uetb


1 
 ue t b 
2
g e tb
However, there are structural stresses, fuel supply and
drag to consider too.
Typical burning times for high-thrust rockets are 30 to
200 seconds
ROCKETS MASS:
Payload (L)  o   L   p   s
 Propellant (P)
 Structure (S, tank and engine)
Payload

tank
engine

Leading to important mass ratios:

o
o

b  L   s

Remaining mass

L
L

o   L  p   s
Payload ratio
 engine   tank
 propellant   engine   tank


s
  L
 b
 p   s o   L
Structural coeff.
1 
 
ROCKETS MASS:
u  u g  D 0


 
 o 
o
 ueq ln 

  ub  ueq ln 
 
 b 


  
Balance  L  o    engine   tank    p 


 
L
 1  engine  tank p  p
o
o
 p o o
 Payload ratio:
Using rocket in space Eq.:   o  eu /u   p  1  eu /u

b

Leading to:
o   p

eq
b
eq
o
 engine /  o   tank /  p

 engine /  o  1   tank /  p  1  e
 u / ueq

Hydrogen performs better, at the cost
of a heavier more complex structure!
MULTISTAGE ROCKETS:
Why bother?
 Each stage can be optimized (weight,
conditions)
 Used stages can be discarded (Drag/weight
reduction)
 Each stage carries the next ones as a
payload

PROPULSION: ROCKET THRUST
Propellant chamber and Nozzle design
OVERVIEW:
Propellants
 Performance :

Characteristic velocity
 Thrust coefficient


Nozzles:
Nozzle shape
 Nozzle length
 Effect of friction
 Effect of Back pressure


Nozzle cooling

Radiation, convection and heat sinks
PROPELLANTS:
In rockets, the most common type are bipropellants,
using two chemicals – a fuel and an oxidizer.
 A tripropellant can take advantage of smaller
molecules getting to a greater exhaust velocity, and
higher propulsive efficiency, at a given temperature.
 Although complex, most designed tripropellant
systems add hydrogen.
 But, hydrogen has a low density and very low boiling
point (Tc=33K) – requires large high pressure, cooled
tanks to liquefy.

SOLID PROPELLANT




Usually a low-explosive material, including diluted high-explosives, in
controlled burning - deflagration not detonation.
Thrust is produced by gas pressure during burning (guns/cannons/rockets).
Common Gun propellants are:
 Gunpowder (black powder), Nitrocellulose-based powder, Cordite,
Ballistite, Smokeless powders
Composite propellants used in rockets are made from :
Solid oxidizer such as ammonium perchlorate or ammonium nitrate
 Rubber binder such as HTPB, or PBAN (or polyglycidyl nitrate or polyvinyl
nitrate for extra energy)
 Optional high-explosive fuels (for extra energy) such as RDX or nitroglycerin




Optionally a powdered metal fuel such as aluminum.
Amateur propellants use Potassium Nitrate, or Potassium perchlorate, with
sugar and epoxy.
Exploding propellants have been experimented with in Pulse Detonation
Engines.
PROPELLANT GRAIN




Solid propellants are porous materials composed of
multiple grains bound together.
A grain is any individual particle of propellant (fuel &
oxidizer) regardless of the size or shape.
Shape, size and composition of a propellant grain
determines the burn time and rate, and therefore the
amount of gas released - thrust vs time profile.
Both the grain (local geometry) and overall propellant
geometry affects burn rate, three types are identified:
Progressive Burn - multiple perforations or center star cut
for large surface area.
 Degressive Burn – outer burning of a cylinder or sphere.
 Neutral Burn – with a single hole, as outside surface
decreases the inside surface increases at the same rate.

PERFORMANCE CHARACTERISTICS:
Assuming: combustion products are an ideal gas,
reaction is equivalent to negligible velocity, constantpressure heating and steady 1D, isentropic expansion
in nozzle
 The thrust process can be described as:

PERFORMANCE CHARACTERISTICS:

The energy equation gives the heat added as:
Q  m  ho 2  ho1   To 2  To1 

qR
cp
Isentropic expansion in the nozzle dictates:
ue2
 ho 2  he  c p To 2  Te 
2

And using the isentropic 1D flow relations:
  p  1/ 
  p  1/ 


ue2
R
q
  c p  To1  R   1   e 


To 2  1   e 


  po 2 


2   1  M
c
p
p 



  o2 

cp
 1 /
ue2  qR    pe 
To 1  qR / c p
     1  

2  M    po 2 






Low molecular weight products give more thrust!
(fuel rich burn)
PERFORMANCE CHARACTERISTICS:

Mass flow of the products can be written as:
 2 
A* po 2
m


RTo 2
  1

 1 /  1
Which leads to a thrust of:

2 2  2 

A* po
  1    1 
po 2

 1 /  1 
p 
1   e 
  po 

 1 /
 p
p A
   e  a  e*
  po po  A

mue
A* Po
Here molecular weight and stagnation temperature
have disappeared – ideal thrust depends only on
pressure!
CHARACTERISTIC VELOCITY:

The characteristic velocity developed in the
combustion chamber:
po A*
1 2 
*
c 

m
    1 

Fuels table:
 1 / 1 
RTo
M
THRUST COEFFICIENT:

The thrust coefficient for a nozzle can be written as:

2 2  2 
C  * 
A po
  1    1 
 1 /  1 
 p 
1   e 
  po 2 

 1 /
 p
p A
   e  a  e*
  po po  A

For an ideal rocket this value is only a function of
nozzle geometry – level of proximity of pe to pa
(conversion of pressure to velocity)
 Therefore, for the entire (ideal) rocket the thrust is:

  mc*C

Comparison of real combustion chambers and nozzles
to the ideal coefficients gives their quality