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AEE 465 / MAE 565: Rocket Propulsion
Prof. Werner J.A. Dahm
Fall 2023 Exam 1
25 October 2023
Open books, notes, homeworks, internet, etc., but you must work alone and you must show all
of your work on every problem to obtain partial credit wherever possible.
On the first page of your exam solutions you must write out this statement, if it applies:
These solutions are entirely my own work; I have not received any help
in any form from any person in solving any parts of this exam.
If this is true, directly below this honor statement you must print and sign your name.
1. A commonly used hypergolic propellant combination is monomethyl hydrazine (MMH) and
nitrogen tetraoxide (NTO). This has been widely used, especially for upper-stage engines and
for attitude control thrusters. Assume these propellants have the following bond structures.
Assume that the overall stoichiometric reaction for these propellants can be approximated as
5
1 CH6 N2 + N2 O4 ! 3.00 H2 O + 2.25 N2 + 0.71 CO2 + 0.59 CO
4
(The H, N, and O atoms are all balanced, and the proportions of the four major species
shown on the product side are correct [relative to the fully detailed equilibrium product
composition from CEA]. Ignore the resulting small C-atom imbalance – because only
1 mole of C atoms is involved in this overall reaction compared to 15.5 moles of other atoms,
namely 6 moles of H atoms, 5 moles of O atoms, and 4.5 moles of N atoms.)
a) Based on this overall stoichiometric reaction, compute the stoichiometric oxidizer-to-fuel
mass ratio for this propellant combination. (5 pts)
b) Using the attached table of bond energies (note both N2 and CO have triple bonds), based
on this overall stoichiometric reaction compute the mass-specific heat of combustion Δhc
for this propellant combination in kJ/kg of fuel. (20 pts)
2. Here you will compare the total ∆V required for two options, shown below, that could be
used for ascent trajectories to reach a desired circular orbit of 500 km above Earth’s surface.
On the left (Case 1), the ascent begins horizontally via an initial engine burn that produces
the needed ∆V1 so that the vehicle ascends along an elliptical trajectory having its perigee at
Earth’s surface and its apogee at the desired circular orbit radius of 500 km above Earth’s
surface. Upon reaching this apogee, a second engine burn produces the needed ∆V2 that
places the vehicle in the desired circular orbit 500 km above Earth’s surface.
On the right (Case 2), the ascent is via an initial engine burn that produces the needed purely
vertical ∆V1 so that the vehicle ascends along an elliptical trajectory having semi-minor axis
b = 0 (thus, a line), its perigee at Earth’s center, and its apogee at the circular orbit radius of
500 km above Earth’s surface. Upon reaching the apogee, a second engine burn produces the
needed ∆V2 that places the vehicle in the desired circular orbit 500 km above Earth’s surface.
Take the Earth’s radius as 6378 km and its gravitational parameter as µ = 3.986 ⋅1014 m3/s2.
For simplicity on this exam, ignore gravity loss and the effect of Earth’s rotation rate.
a) For Case 1, calculate e for this elliptical ascent trajectory. (5 pts)
b) For Case 1, calculate a for this elliptical ascent trajectory. (5 pts)
c) For Case 1, calculate the needed ∆V1. (5 pts)
d) For Case 1, calculate the needed ∆V2 and the resulting ∆Vtotal. (5 pts)
e) For Case 2, calculate e for this “elliptical” ascent trajectory. (5 pts)
f) For Case 2, calculate a for this “elliptical” ascent trajectory. (5 pts)
g) For Case 2, calculate the needed ∆V1. Note in this extreme case the usual equations for
elliptical trajectories are not easy to apply, so … instead use a more fundamental equation
from which we obtained the “usual” equations for elliptic trajectories. (5 pts)
h) For Case 2, calculate the needed ∆V2 and the resulting ∆Vtotal. (5 pts)
i) Which of these two cases requires the smaller ∆Vtotal ? (2 pts)
3. In this problem, you will compare (1) a single-stage-to-orbit (SSTO) launch vehicle and (2) a
two-stage-to-orbit (TSTO) launch vehicle for placing a 900 kg (~1 ton) final payload mass
into a circular orbit 300 km above Earth’s surface using an elliptical ascent trajectory such as
that shown below.
The two launch vehicles (SSTO and TSTO) each provide only the first velocity increment
∆VA that places the payload into the proper elliptical ascent trajectory. (The payload has its
own rocket engine that later provides the second velocity increment ∆VB once the payload
has reached the apogee of the elliptical ascent trajectory at the desired circular orbit height.)
If you were to follow the same procedure as in Problem 2 (you do not need to do this) you
would find that the needed first velocity increment is ∆VA = 7996 m/s. Thus, the SSTO and
TSTO launch vehicles each need to produce this value of ∆V.
For simplicity on this exam, assume:
• All stages on both launch vehicles have the same structural coefficient e = 0.090.
• The two stages on the TSTO launch vehicle have the same propellant mass ratio R.
• Each vehicle’s propulsion system produces the same equivalent velocity Veq = 4970 m/s.
• Ignore the effect of Earth’s rotation rate in the ∆V budget.
a) First consider the SSTO launch vehicle. Because the engine burn that produces ∆VA is
oriented horizontally and thus gravity losses are zero, we can apply the Tsiolkovsky
rocket equation to this problem. Use it with the information given above to determine the
propellant mass ratio R for the SSTO launch vehicle. (5 pts)
b) From the resulting propellant mass ratio R and the given structural coefficient e,
determine the payload ratio l for the SSTO launch vehicle. (5 pts)
c) Use these values to determine the propellant mass MP, the structural mass MS, the
payload mass ML (given), and the total mass M0 for the SSTO launch vehicle. (5 pts)
d) Now consider the TSTO launch vehicle. Apply the Tsiolkovsky rocket equation to this
N = 2 rocket to determine the propellant mass ratio R for each of the two stages (recall we
are taking the two stages to have the same propellant mass ratio). (5 pts)
e) From this propellant mass ratio R and the given structural coefficient e, determine the
payload ratio l for each of the two stages of the SSTO launch vehicle. (5 pts)
f) Use these values to determine the propellant mass MP, the structural mass MS, the
payload mass ML (given), and the total mass M0 for the TSTO launch vehicle. (5 pts)
g) How closely does the payload ratio from part (e) compare with the optimal payload ratio
based on the given final payload mass and the total mass M0 from part (f) for the TSTO
launch vehicle? (3 pts)
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