Mission Configuration – 100g Payload – Lunar Descent
Section A – 5.4.3
A – 5.4.3 – Propulsion
Propellant and Propulsion System Selection
To select the Lunar Lander Propulsion system, we began with a preliminary survey of the
possible propulsion system types. Included in this survey are mono–propellant, bi–
propellant, hybrid and solid propellant systems. Key items of interest are the particular
system Isp, reliability, availability and cost. Starting with the ideal rocket equation it is
possible to derive an equation that computes the required propellant mass for a given
propulsion systems specific impulse, Isp.(Humble, 1995)
(𝑒 (∆𝑉/𝐼𝑠𝑝∙𝑔𝑜 ) −1)
𝑚𝑝𝑟𝑜𝑝 = (𝑚𝑝𝑎𝑦 1−𝑓
𝑖𝑛𝑒𝑟𝑡 𝑒
(∆𝑉/𝐼𝑠𝑝∙𝑔𝑜 )
(A – 5.4.3–1)
(1 − 𝑓𝑖𝑛𝑒𝑟𝑡 ))
By assuming a payload mass, mpay, of 85 kg, minimum and maximum finert values of 0.1
and 0.25 (based off of historical data) and a ∆V of 1950 m/s, we use Eq. (A – 5.4.3–1) to
compute the required propellant mass as a function of the propulsion system Isp.
240
220
200
Propellant mass [kg]
180
160
Mono-Prop
140
Hybrid
120
100
80
60
Bi-Prop
40
20
150
200
250
300
350
400
450
500
550
Isp [s]
Fig. A – 5.4.3–1 Propellant mass required vs. propulsion system Isp
(Thaddaeus Halsmer)
Author: Thaddaeus Halsmer
600
Mission Configuration – 100g Payload – Lunar Descent
Section A – 5.4.3
In Fig. A – 5.4.3–1 the two dashed lines represent the required propellant mass versus Isp
for the maximum and minimum finert values, where finert is the inert mass of the space craft
excluding propellant and payload divided by the sum of the inert mass of the space craft
and mass of the propellant. Because the finert values are chosen as historical limits, when
we plot data points for various propulsion system’s empirical Isp values, they fall within
this envelope. The result of propellant mass versus Isp study shows us that an exponential
propellant mass penalty is paid for choosing a propulsion system with a lower Isp.
Assuming that Lunar Descent should conclude with a fully controlled soft landing leads
us to the requirement of an engine with variable thrust. Varying the thrust of a rocket
engine is achieved by changing the mass flow rate of propellants as seen in the vacuum
thrust equation
𝑇ℎ𝑟𝑢𝑠𝑡𝑣𝑎𝑐 = 𝐼𝑠𝑝,𝑣𝑎𝑐 ∙ 𝑚̇ ∙ 𝑔𝑜
where Isp is assumed constant, 𝑚̇ is the propellant mass flow rate and go is the Earth
gravity acceleration constant. We initially consider mono–propellant, bi–propellant and
hybrid propulsion systems because they fulfill this requirement by varying the propellant
mass flow rates. Both mono–propellant and hybrid propulsions systems have the added
advantage of being throttled by controlling only a single fluid mass flow rate. In a bi–
propellant engine both oxidizer and fuel flow rates must be controlled simultaneously to
maintain the correct oxidizer to fuel ratio, O/F.
An additional constraint on the propulsion system selection is that there are currently no
commercially available rocket engines sold that meet the variable thrust requirement and
scale needed for our mission.
This makes the cost of development of the chosen
propulsion system another significant factor. The bi–propellant option is eliminated as a
possibility because of the complexity of bi–propellant rocket engines and their associated
feed systems. It is not realistic to suggest that such a system could be developed with
sufficient reliability within the time and cost constraints of this project.
Author: Thaddaeus Halsmer
Mission Configuration – 100g Payload – Lunar Descent
Section A – 5.4.3
Based on expert advice it was determined that an H2O2 / Polyethylene Radial Flow
Hybrid Engine (RFHE) propulsion system of appropriate scale could be developed within
one year and for less than $250000.
(Heister, 2009)
Using CEA we determined that this
system could easily achieve a vacuum Isp > 300 s.(Gordon, 1971) Based on this information
and the relative propellant masses, shown in Fig. A – 5.4.3–1, we selected the H2O2 /
Polyethylene hybrid propulsion system over the mono–propellant system because of its
significant propellant mass savings and low cost of development.
Radial Flow Hybrid Rocket Engine Overview
Traditional hybrid rocket engines contain a solid fuel grain in the combustion chamber
with liquid oxidizer being injected into axial ports in the fuel grain. However, this
configuration leads to engines with a high L/D which is not well suited to our compact
landing vehicle. Because of the dimensional constraints we chose an experimentally
proven radial flow configuration that allows the combustion chamber to have a
significantly larger diameter and shorter length.
Fig. A – 5.4.3–2 Radial Flow Hybrid engine configuration
(Thaddaeus Halsmer)
Figure A – 5.4.3–2 presents the architecture of a radial flow hybrid rocket engine. It also
shows how the oxidizer is injected around the combustion chamber perimeter between
the solid fuel grains, with the combustion gases exiting through a hole in the lower fuel
grain into the nozzle.
Author: Thaddaeus Halsmer
Mission Configuration – 100g Payload – Lunar Descent
Section A – 5.4.3
Chamber Pressure & Feed System Selection
Space engines have the benefit of being able to achieve high efficiency at relatively low
chamber pressures. As a result the entire oxidizer feed system mass can be reduced
because of the lower operating pressures. We used CEA to perform a preliminary Isp
versus chamber pressure sensitivity study and found that the benefits of reducing the
chamber pressure outweighed the minimal change in Isp.
A key assumption that was made in determining the chamber pressure operating range is
that the engine would be designed for a 10:1 throttle ratio. Deep throttling to this degree
has experimentally been proven to be possible without extraordinary difficulty.(Sutton, 2001)
Because the chamber pressure of the engine changes by almost exactly the same factor as
thrust, we assume that the change in chamber pressure is 10:1 for an engine with a 10:1
throttle ratio. Based on this assumption the maximum chamber pressure is 10 times the
minimum possible chamber pressure at which steady combustion is maintained. While
the exact minimum chamber pressure must be experimentally determined for this
particular design, hybrid engines using the same propellants have demonstrated steady
operation at chamber pressures as low as 0.21 MPa. Therefore 0.21 MPa is the assumed
minimum chamber pressure for this design, making the maximum steady state chamber
pressure 2.1 MPa. Based on these pressures the optimum feed system architecture is a
stored gas–pressurization configuration.
Mean Isp and Combustion Chamber Sizing
Sizing of the combustion chamber for the RFHE is based on the dimensions of the fuel
grains shown in Fig. A – 5.4.3–2. The fuel grain dimensions are dictated by the fuel
grain regression rate, the desired oxidizer to fuel ratio and thrust. Hybrid engine fuel
regression rates can be modeled using the following equation
𝑟 = 𝑎𝐺𝑜𝑛
Author: Thaddaeus Halsmer
(A – 5.4.3–2)
Mission Configuration – 100g Payload – Lunar Descent
Section A – 5.4.3
where r is the fuel regression rate in m/s, Go is the total oxidizer mass flux in kg/m2–s,
and a and n are empirically derived constants. Multiplying Eq. (A – 5.4.3–2) by the fuel
grain density ρf and the burn area Ab, the equation for the fuel mass flow rate is derived.
𝑚̇𝑓 = 𝑎𝐺𝑜𝑛 𝜌𝐴𝑏
(A – 5.4.3–3)
Fig. A – 5.4.3–3 Fuel grain dimension definitions
(Thaddaeus Halsmer)
Using the variable definitions in Fig. A – 5.4.3–3 along with the definition of Go,
substituting them into Eq. (A – 5.4.3–3) yields an expression for the fuel mass flow rate
as a function of the fuel grain/combustion chamber geometry
𝑚̇𝑓 = 𝑎 (
𝑚̇𝐻2𝑂2 𝑛
2𝜋𝑟ℎ
) 𝜌𝑓 𝜋(2𝑟𝑔2 − 𝑟𝑝2 )
(A – 5.4.3–4)
By assuming that all terms in Eq. (A – 5.4.3–4) are constant except h, which is changing
at a rate proportional to the fuel regression rate r, we can derive the oxidizer to fuel ratio,
O/F as a function of h
𝑂/𝐹 = 𝑘(ℎ)𝑛
(A – 5.4.3–5)
where k is a empirically derived constant and n is a constant that has been theoretically
shown to be approximately 0.8. From Eq. (A – 5.4.3–5) we can see that the O/F ratio
will increase with time and we know that a change in the O/F changes the engines Isp.
Maximum Isp occurs near the stoichiometric O/F making it is necessary to size the fuel
Author: Thaddaeus Halsmer
Mission Configuration – 100g Payload – Lunar Descent
Section A – 5.4.3
plates so that initially the engine is fuel rich and as O/F increases the mean Isp is
maximized.
Using CEA integrated into a Matlab script, along with an empirical fuel regression rate of
0.000559 m/s
Caravella
and Eq. (A – 5.4.3–5) it is possible to iteratively find an initial O/F
ratio that maximizes the mean Isp over the burn time. Figure A – 5.4.3–4 is an example of
the program output, over a 200 s burn time, and it demonstrates the significant O/F shift
along with the corresponding changes in Isp.
340
330
320
Isp [s]
310
300
290
280
270
260
0
2
4
6
8
10
12
14
16
18
20
O/F ratio
Fig. A – 5.4.3–4 Isp vs. O/F for 200s burn time
(Thaddaeus Halsmer)
Through this approach, we can determine the initial dimensions of the fuel grains and
from these the dimensions of the combustion chamber are derived.
Nozzle Sizing Analysis
For rocket engines operating in space, as the nozzle area ratio A exit/Athroat is increased, the
engines Isp also increases. However, the cost of increasing Isp with a increase in nozzle
area ratio is the added mass and size of the physically larger nozzle. Because an increase
in Isp reduces the required propellant mass, there theoretically exists a optimum nozzle
mass and corresponding area ratio. To make a preliminary examination of the tradeoff
between nozzle area ratio and the corresponding propellant mass, a Matlab script was
written that uses CEA to compute the Isp of the engine for a given area ratio along with
the corresponding nozzle mass and propellant mass. Thrust and burn time are kept
Author: Thaddaeus Halsmer
Mission Configuration – 100g Payload – Lunar Descent
Section A – 5.4.3
constant and only the nozzle area ratio is varied. The mass of the nozzle is determined
using an empirically derived equation that estimates the mass of a carbon phenolic hybrid
engine nozzle.(Humble, 1995)
Inputs for this equation are the nozzle area ratio 𝜀, and total
propellant mass mprop.
Nozzle + Prop mass [kg]
𝑀𝑛𝑜𝑧𝑧𝑙𝑒
𝑚𝑝𝑟𝑜𝑝 2⁄3 𝜀 1⁄4
= 125 (
) ( )
5400
10
128
Pc = 1.72 MPa, 2000N thrust
127.5
127
126.5
126
50
100
150
200
nozzle area ratio 
Fig. A – 5.4.3–5 Mass penalty vs. nozzle area ratio trade study result
(Thaddaeus Halsmer)
Figure A – 5.4.3–5 is an example of the output generated by the script. It shows that the
minimum combined mass of the nozzle and propellant occurs at an area ratio of
approximately 150. Based on this analysis we chose an area ratio of 100, because there is
only a minimal mass penalty and this nozzle’s dimensions are near the maximum that can
be integrated into the vehicle.
We compute the throat area of the nozzle as a function of the engine thrust F, coefficient
of thrust Cf, and the chamber pressure Po using the following equation
𝐴𝑡 = 𝜉
𝐹
𝑓 𝐶𝑓 𝑃𝑜
Author: Thaddaeus Halsmer
(A – 5.4.3–6)
Mission Configuration – 100g Payload – Lunar Descent
Section A – 5.4.3
where Cf is computed by CEA and the thrust correction factor 𝜉𝑓 is assumed to be 0.96.
(Sutton, 2001)
Once the nozzle area ratio and throat diameter are known, the exit diameter is
easily computed. Finally, we can estimate the length of the nozzle using an empirically
derived expression for the length of 80% bell contour nozzles
𝐿𝑛𝑜𝑧𝑧𝑙𝑒 = 0.739𝐷𝑡 ∙ 𝜀 0.643
where Dt is the throat diameter of the nozzle.
Author: Thaddaeus Halsmer
(A – 5.4.3–7)