Download 4262: Rockets and Mission Analysis

4262: Rockets and Mission Analysis
Homework #4
Assigned: October 16, 2014
Due: October 28, 2014
The purpose of this homework assignment is to explore the implications of electric propulsion
and optimum exhaust velocity.
Question 1
An electrostatic rocket uses heavy particles for a propellant with a charge-to-mass ratio of 500
C/kg to produce a specific impulse of 3,000 seconds.
a. What acceleration voltage would be necessary?
b. With one-dimensional space-charge-limited current and a maximum allowable gradient
of 105 V/cm, what is the diameter of a round beam producing 0.5 N of thrust?
c. With a charge to mass ratio of 500 C/kg, the required acceleration voltage is about 9x105
Volts (answer to part (a)). What are your thoughts on this number? Is it prohibitively
large? The problem states, “An electrostatic rocket uses heavy particles…”. If the ions
are singly charged, how heavy are these particles in kg?
Question 2
A cesium ion rocket is to be used at a specific impulse of 5,000 seconds. Assume the cesium to
be 100% singly ionized (q/m = 7.25 x 105 C/kg).
a. What is the required acceleration voltage and the bean power per unit thrust?
b. If the cesium is 90% singly ionized and 10% doubly ionized, what is the specific impulse
and bean power per unit thrust, assuming that the acceleration voltage is unchanged?
Question 3
A resistojet is to be designed to supply a specific impulse of 310 seconds and a total thrust of 10
N. It is to be operated with ammonia in a nozzle with an area ratio of 100:1. The specific heat
ratio during the expansion can be estimated at 1.3, and the expansion may be assumed to be
adiabatic. Determine the maximum temperature for the following two cases:
a. No dissociation of the NH3
b. 50% dissociation of the NH3 (into N2 and H2) in the heating chamber, following by a
frozen (non-changing) expansion in the nozzle.
Question 4
If the energy is available in electrical form, then there is no limit in principle on the exit velocity,
other than the speed of light, for a vehicle. In practice, it is indeed true, that we can achieve very
high exit velocities with good efficiency by using any of a number of electrical accelerators. For
electrical propulsion, it is important to take into account the mass of the energy source, which
may be a large fraction of the total mass of the rocket system. We can write the total mass as the
sum of the payload, the propellant, the engine, the structure, and the electrical conversion
equipment:
M tot  M pay  M prop  M eng  M struct  M elec
1
1
In this example, where we are interested in the V we can impart to a given payload, we will
neglect the mass of the engine and structure relative to the mass of the electrical conversion
equipment. We also define the specific mass, or the mass-to-power ratio, of the electrical
conversion equipment as:

2
M elec
P
Making use of the assumptions provided, as well as taking into account the initial acceleration of
the rocket, ao, show the ratio of the payload mass to the total mass can be expressed as:
M pay
M total
  V 

Ve 
 e
 a V  Ve 
 o


 2  V 
3
Now for a given V, ao and , we want to choose the Ve to maximize our expression for
To do so differentiate equation 3 with respect to the non-dimensional group
equal to zero, and solve for the optimum
M pay
M total
.
V
, set the result
Ve
V
. When taking the derivative, you may treat the
Ve
  a V 
quantity  o
 as a constant.
 2 
M pay
V
  a V 
Plot the quantity  o
versus optimum
. Comment on these results. In
 and
Ve
M total
 2 
V
particular, comment on the
bound that should be chosen. Is there an upper limit on
Ve
  a o V 

 that makes physical sense for a given mission? Does this upper limit coincide with the
 2 
maximum value?
Next consider an example for a mission that requires V=104 m/s and a technology that enables
M pay
=0.02 kg/W. How long would a mission take if we prescribe
=0.5?
M total
2