Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
A-5.3.3 - Propulsion on the Orbital Transfer Vehicle
Propulsion System Sizing
Traditional methods for propulsion sizing are based on chemical propulsion - one could
work with estimates for the inert mass fraction and delta V and then use the standard
forms of the ideal rocket equation to obtain required propellant masses. For EP, however,
propellant mass is only part the story; the required power is actually the true driver of the
system. It is therefore necessary to work with a new set of basic sizing equations that,
although still derived out of the ideal rocket equation, account for the power system of
the spacecraft. This section describes the processes and equations we use.
The very first step in the sizing process is to clarify what requirements we have to build
on. From the Lunar Transfer perspective, the payload is a fixed input, comprised of the
wet mass of the landing and roving vehicles. Using a guess for an initial mass and
selecting an arbitrary time of flight, the trajectory code produces a thrust requirement for
the OTV. So in general, the problem takes the following form:
Given:
Find:
1)
Payload mass
2)
Thrust
3)
Time of flight
1) Required power
2)
Propellant Mass
3)
Inert system mass
A set of secondary constraints further define the design space. This includes physical
limits to what the launch vehicle can support, or a finite budget for the project. We also
use a few simplifying assumptions to make the analysis manageable. These items are
listed below.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
Constraints: 1) System must fit inside launch vehicle payload fairing
2)
The best solution is the cheapest solution
3)
Maximum time of flight is 1 year  System must be ready to
launch by Dec. 31st, 2011
Assumptions: 1)
2)
The thruster operates with a constant mass flow rate
The specific power of the spacecraft will be similar to historical
missions
The second assumption is necessary to begin the design process; however it is relaxed
once we know more details of the spacecraft components.
The required thrust inherently defines a required mass flow rate and specific impulse,
although in no particular combination. Equation A-5.3.3-1 is the most basic form of the
thrust equation. Pressure effects in EP are insignificant and ignored.
𝑇 = 𝑚̇𝐼𝑠𝑝 𝑔0
(A-5.3.3-1)
where T is the thrust, 𝑚̇ is the mass flow rate, 𝐼𝑠𝑝 is the specific impulse, and 𝑔0 is the
acceleration due to gravity. The above equation is simple, but it is the most critical
relationship in defining the propulsion system and the overall spacecraft. Multiplying the
mass flow rate by the time of flight gives the total propellant mass. For a time of flight as
large as one year, a small change in 𝑚̇ has big implications for the initial mass of the
OTV. On the other hand, the specific impulse is a direct function of input power, which
in turn has drastic effects on the size of solar panels needed.
A brief explanation of how an EP system generates thrust helps the understanding of the
specific impulse / input power relationship. A Hall Thruster fundamentally operates by
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
ionizing the propellant to a plasma state and using an electrostatic field to accelerate the
ions out of the chamber. An applied magnetic field is required to direct the ions along a
closed path and to tweak the fields for optimum efficiency and lifetime. The stronger
these fields are, the faster the Xenon ions are accelerated, and the higher the specific
impulse. Once materials and geometry are set, the only way to increase the strength of the
electric fields is to increase the input power. The required power for the HET is given by
Eq. A-5.3.3-2. Note that the power is a linear function of mass flow rate, but more
importantly a quadratic function of exhaust velocity (or Isp).
𝑃𝑟𝑒𝑞 =
1
𝑚̇𝑣𝑒2
2
𝜂
(A-5.3.3-2)
where 𝑃𝑟𝑒𝑞 is the required power, 𝑣𝑒 is the exhaust velocity, and η is the thrust efficiency.
So, increasing the mass flow rate will increase the spacecraft mass and increasing the
specific impulse will increase the power requirement. We find an optimum combination
of these two parameters by considering the cost of the mission. Xenon propellant
currently sells for approximately $1200 per kg(Spores(2005)), and the solar arrays are
purchased at a price of $1000 per Watt. The extra propellant or solar array mass penalty
is compounded because of the launch cost, which for the Dnepr is estimated at $3400 per
kg. This basic relationship is summarized in Eq. A-5.3.3-3. Here we note that total cost
does not actually include the full price of the mission, just a total useful for propulsion
sizing.
𝐶𝑡𝑜𝑡𝑎𝑙 = 𝑃𝑟𝑒𝑞 𝑐𝑃 + 𝑚𝑋𝑒 𝑐𝑋𝑒 + 𝑚0 𝑐𝐿𝑉
(A-5.3.3-3)
where 𝐶𝑡𝑜𝑡𝑎𝑙 is the total cost, cp is the price-per-watt of solar cells, cXe is the price-perkilogram of Xenon, and cLV is the price-per-kilogram to use the launch vehicle. There are
two paths to take to obtain the initial mass m0. In the first method, we assume a historical
value for the specific power, α, of the entire propulsion system. Careful definitions are
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
important for the specific power, which is the total generated power dedicated for EP (in
this case by the solar arrays) divided by the mass of the propulsion system (units of
W/kg). The mass of the propulsion system includes the traditional items such as the
thruster and tank, in addition to power-related items such as the PDCU and PPU.
Historical values of α fall between 30 and 50 W/kg. Using this specific power estimate,
we use Eq. A-5.3.3-4 to approximate the inert propulsion system mass (Turchi).
𝑚𝑖𝑛𝑒𝑟𝑡 =
𝑣𝑒2 𝑚𝑋𝑒
2𝛼𝜂𝜏
(A-5.3.3-4)
where minert is the inert mass of the propulsion system, α is the specific power of the
system, and τ is the time of flight. The jet efficiency, η, is another performance parameter
of the Hall Thruster, distinct but related to specific impulse. Conceptually, jet efficiency
is the ratio of flow power (12 𝑚̇𝑣𝑒2 ) to thrust. We find values for the efficiency from
empirical curves, which are explained later in this section.
Once the inert mass is calculated, the initial mass is found as the sum of the propellant
mass, inert mass, and payload mass. Then we calculate the total cost in Eq. A-5.3.3-3.
After iterating through several values for mass flow rate, an optimum (minimum cost)
solution is found.
This cost analysis reveals the interesting trend that the price of the solar cells is the single
largest driver in the propulsion system design. For the EP-based Orbit Transfer Vehicle, a
system optimized on cost is very different from a system optimized on mass. Consider
Figure A-5.3.3-1. Here the cost of the system, calculated using Equation A-5.3.3-3, is
plotted for different values for time of flight. Each time of flight has an associated
required thrust, per the trajectory code. This data set points to the optimum time of flight
for our mission.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
Mission Cost vs. Time of Flight
18,000,000
16,000,000
Cost (US $)
14,000,000
12,000,000
10,000,000
8,000,000
Solar Cost = $1000 / W
6,000,000
Solar Cost = $100 / W
4,000,000
2,000,000
0
0
100
200
300
Time of Flight (days)
400
Figure A-5.3.3-1: The propulsion system cost versus time of flight for two different prices of
solar cells.
(Brad Appel)
If the solar cells could be obtained for $100 / W, our mission time of flight would be
eight months. But at $1000 / W, which is the team’s best quote, the optimum time of
flight is one year. Essentially, the longer mission is the cheaper mission, but that fact is
highly sensitive to the price of solar arrays.
Thruster Selection
The second method for finding inert mass involves calculating the masses of the
individual spacecraft components rather than bulking everything together into Eq. A5.3.3-4. We use this more accurate method as specific data on other subsystems become
available. The sizing scheme is iterative on the mass flow rate and specific impulse
combination. In general, the sizes of many of the spacecraft components are dependent
on those iterating variables. We therefore find it more realistic to incorporate empirical
curve-fit data. Within each iteration, the inert mass is calculated as a function of either
mass flow rate, specific impulse, or power (whichever data is available). We find
empirical data from journal articles or company websites. The curve-fit for the propellant
tank is shown in Fig. A-5.3.3-2. The data come from the SMART-1, Deep Space 1, and
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
Dawn missions, as well as commercial tanks from Pressure Systems Inc. and EADS
Launch Vehicles(Brophy).
Tank Dry Mass [kg]
Tank Mass Curvefit (Low Mass Systems)
60
50
40
30
20
y = 6E-09x4 - 8E-06x3 + 0.0031x2 - 0.2887x + 14.083
R² = 0.9997
10
0
0
100
200
300
Xenon Mass [kg]
400
500
Figure A-5.3.3-2: Historical data for Tank mass as a function of Xenon mass. Only low-mass
missions were considered.
Empirical curves are used for computing the jet efficiency and other performance
characteristics as well(King),(Szabo (2005)),(Szabo(2007)). Examples of these data are presented in
the Figs. A-5.3.3-3 and A-5.3.3-4.
BPT-2000 Isp and Efficiency vs. Power
Jet Efficiency
0.6
0.55
y = -3E-21x2 + 3E-05x + 0.4286
R² = 1
0.5
0.45
0.4
0
500
1000
1500
2000
Power (W)
2500
3000
Figure A-5.3.3-3: Efficiency curve-fit for the Aerojet BPT-2000(King).
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
Power (W) and Efficiency
SPAD Hall Thruster Trends
0.8
0.7
0.6
0.5
0.4
0.3
y = -4E-08x2 + 0.0003x + 0.0492
R² = 0.9894
0.2
0.1
0
0
1000
2000
Isp (s)
3000
4000
Figure A-5.3.3-4: Efficiency for empirical data out of Space Propulsion
Analysis and Design, chapter 9. (Turchi).
The empirical data plays a role to help decide which of the available Hall Thrusters is
best suited for our mission. By running the optimization scheme for each set of empirical
data, we see which thruster will provide the cheapest or lightest mission. Figure A-5.3.35 shows the result of this analysis. The total mission mass is broken down into
subsystems.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
Initial Mass Breakdown by Thruster Choice
800
700
Injected Mass to LEO [kg]
600
Contingency
Plumbing
Thruster
PPU
Solar Array
Attitude
Tank
Xenon
Structure
Payload
500
400
300
200
100
0
SPAD BPT-2000 MELCO BHT-1500
Figure A-5.3.3-5: Mass breakdown for the OTV when different thrusters are used. NOTE:
This is not based on the team's final configuration data.
(Brad Appel)
Note that the “SPAD” bar uses historical data in the textbook Space Propulsion Analysis
and Design; we include this as a sanity check. We conclude from the mass breakdown
that the BHT-1500 thruster is the best option. Quotes were not available for all the Hall
Thrusters, but we do know the BHT-1500 is substantially cheaper than the BPT-2000.
Why Hall Thrusters?
There are many types of electric propulsion thrusters, and each could provide a
significant payload advantage over chemical propulsion. However, in terms of both high
specific impulse and technological maturity, there are really only two options: an ion
thruster or a hall thruster. An ion thruster is actually capable of higher specific impulses
than our HET, however it operates at a much lower specific power – almost four times
lower for our mission.(Spores) This means that to accomplish the same mission, the ion
thruster would require nearly four times as much power. Since we’ve established that the
price of solar power is an enormous driver in mission cost, we decided that ion thrusters
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
are not too practical. If the mission had more demanding requirements, such as a specific
impulse closer to 3000s rather than 2000s, the ion thruster becomes more feasible.
A common concern with Hall Thrusters is erosion of the electrodes. With a time of flight
of one year, we require that our HET be operational for 8760 hours, which is a tight
requirement to be sure. Lifetime tests on Hall Thrusters for times greater than 5000 hours
are very rare. However, a combination of empirical curves and prediction methods both
show the trend that the electrode lifetime drastically increases as we reduce the power
input (this makes sense since we would then be discharging less current to the chamber
walls). A paper by W. Ethan Eagle in 2008, “The Erosion Prediction Impact on Current
Hall Thruster Model Development” describes a method which suggests that our type of
Hall Thruster at our input power (about 1500 Watts) could be operated for around 10,000
hours. So the lifetime capability of the Hall Thruster is close, but because we are
operating it near the bottom of its power operating range, it should provide enough
margin.
Notes on Optimization
The sizing algorithm we use ignores the fact that an optimum specific impulse exists for
every EP mission. Equation A-5.3.3-5 is an expression for the system payload mass ratio
(MPL/Mo) as a function of the delta V, specific impulse, specific power α, thrust
efficiency η, and the time of flight tb.
𝑀𝑃𝐿
𝑀0
= 𝑒
𝛥𝑉
𝐼𝑠𝑝 𝑔0
−
−
(𝐼𝑠𝑝 𝑔0 )2
2𝛼𝜂𝑡𝑏
(1 − 𝑒
𝛥𝑉
𝐼𝑠𝑝 𝑔0
−
)
(A-5.3.3-5)
Solutions to this equation are plotted for various assumed values of the delta V in Fig. A5.3.3-6. To optimize the payload ratio, the ideal specific impulse is 2850 seconds.
However, the optimization curve is not particularly steep, and range of 1800 – 4250
seconds can be accepted with only a 6% loss from the optimum payload ratio.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
Optimum Isp
0.7
0.6
pl
Payload Ratio, M /M
0
0.5
0.4
0.3
0.2
0.1
Delta V = 6 km/s
Delta V = 8 km/s
Delta V = 10 km/s
0
-0.1
1
2
3
4
5
6
Specific Impulse [s]
7
8
9
10
4
x 10
Figure A-5.3.3-5: Payload ratio as a function specific impulse, for a generic mission.
Some clarification is needed to explain why our “optimum solution” is significantly
different from the theoretical “optimal solution.” Basically, the optimum case predicted
by Equation A5.3.3-5 completely ignores the fact that the price of power on our
spacecraft is much higher than the price of Xenon propellant. The “theoretical” optimum
solution assumes we can tack on as many solar cells as we want with no consequences.
Since we are using a “practical” optimum solution, we know that it is a better idea to
trade more propellant mass for less power.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
Power Processing Unit
Power input is required for four primary components: an annular anode, a thermionic
hollow cathode, a neutralizing ion source, and three or more outer magnetic coils. The
PPU runs one cable to the thruster which contains all of these connections, so no extra
breakout harness is necessary.
As part of a feasibility study, we necessarily draw a line between what level of analysis is
fundamentally important and “game-changing”, and what analysis should really be saved
for a more detailed design effort. This means understanding which parts of the system are
set in stone when you buy it off the shelf, and which parameters are open (and useful) for
the customer to optimize. With regards to the Power Processing Unit, we limit our
analysis to looking at historical performance numbers. We lack details of the particular
PPU that Busek would integrate with the BHT-1500. The specifications listed for the
PPU in Table A-5.3.3-3 come from a combination of industry-average values found in
journal articles rather than a specific model (sources included in Mission Architecture).
The most important number for the PPU is its efficiency, which we set at 93% for our
analysis.
Tank Selection
Xenon becomes supercritical at a very specific temperature and pressure. The blue line in
phase diagram in Fig. A-5.3.3-6 shows the division between the liquid and gaseous state
for Xenon. The red line represents the path the Xenon will take during our mission.
Property data come from the National Institute for Standards and Technology website.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
Temperature (K)
Figure A-5.3.3-6: Phase diagram of Xenon including the supercritical state.
Above and to the left of the light blue line is the liquid phase. Because it would
completely invalidate the performance of the Flow Control System, the Xenon must
always be kept at a temperature warm enough to behave as a gas. Note that the propellant
will not stay in the supercritical state throughout the whole mission.
To be both light-weight and high-strength, the tank needs to be made out of advanced
materials that cannot be processed in house. Aluminum, for example, could do the job but
the tank would weigh more than what other materials could achieve. Consider the
following tank sizing analysis.
The total mass of Xenon required for our GLXP missions is about 150 kg. Stored
supercritically at 150 bar (2200 psi), Xenon has a density of about 1600 kg/m3. These
conditions require a propellant volume of 0.094 m3 (5720 in3). We assume the tank is a
sphere, the material is Aluminum 6061-T6551 (yield stress σ is 40 ksi), and a factor of
safety FS of 1.5. With our propellant mass, the spherical radius r is 11 inches. With all of
this information, we solve Equation A-5.3.3-6 for the required thickness t of the tank
wall. The pressure P is the proof pressure of the tank, which we set to 3000 psia.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
𝑡=
𝐹𝑆•𝑃𝑟
2𝜎
Section A-5.3.3
(A-5.3.3-6)
The resulting wall thickness is 0.056 inches. Multiplying by the tank surface area and
density, we come up with a tank mass of 18 kg.
Composite-overwrapped tanks are commercially available which could hold the same
pressure and volume with a dry mass of 10 to 15 kg(Tam). Although we expect them to be
slightly more expensive than the cost of manufacturing the Aluminum tank ourselves, we
choose the commercial tanks for the added reliability. Companies who manufacture such
tanks include ATK Space Systems, Arde Inc., and Lincoln Composites.
Flow Control System
Separate propellant flow is required for three components within the HET: the anode,
cathode, and neutralizer. The anode takes in about 95% of the mass flow rate. The
neutralizer generates a stream of electrons which enter the exhaust plume and cancel out
the positive charge of the Xenon ions. Without this feature, part of the exhaust plume
would accelerate back towards the thruster and cause a serious net charge on the
spacecraft.
We design the Flow Control System to accomplish three main tasks:
1) Isolate the high-pressure storage tank
2) Regulate the pressure down to an operating level
3) Regulate the mass flow rate entering the thruster
The assembly consisting of the solenoid latch valve, pressure regulator, high-purity filter,
sintered flow restrictors, and various pressure transducers shown in Section 5.3.2
demonstrate the bare minimum of parts necessary for the flow to function. Most likely,
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
we would add redundant lines to the system for the real mission. There would also need
to be a series of check, vent, and fill valves, but their masses and cost are low enough to
be neglected. The general layout of the system is adapted from several papers describing
flight hardware(Ganapathi),(Bushway
III).
We also had very helpful contact with Edward
Bushway from Moog Inc.
Pressure Drop
Near the end of the mission life, the tank storage pressure will have dropped drastically.
Because the FCS requires a minimum pressure difference to function, we need to
investigate where the tank pressure will be after 365 days.
First we develop an estimate for the pressure drop in the FCS. Table A-5.3.3-1 lists the
major components in the FCS and their pressure drop according to the manufacturer
. The pressure regulator “locks-up” below 50 psia. Because our design is conceptual
(Moog)
only, we ignore the need for some redundancies in isolation. To account for this, the
pressure drops of the main components are counted twice.
Table A-5.3.3-1 Flow Control System Pressure Drop
Part
Latch Valve
Regulator
Filter
Flow Restrictor
Feed Lines
Redundancy
Total
Pressure Drop
1
50
2
40
2
3
98
Units
psid
psid
psid
psid
psid
psid
psid
So the Xenon won’t make it through the Flow Control System if the tank pressure drops
below 98 psia. We calculate the pressure drop in the tank using the equation of state,
shown in Eq. A-5.3.3-7. The gas constant for Xenon is 63.3 J/kgK.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
𝑚
𝛥𝑃 = (𝛥 𝑉 )𝑅𝑇
Section A-5.3.3
(A-5.3.2-7)
where 𝛥𝑃 is the pressure drop, Δm is the change in Xenon mass, V is the tank volume, R
is the gas constant, and T is the temperature. We know that the temperature of the tank
isn’t changing since we specifically added a thermostat for that reason. Also, of course,
the volume of the tank remains constant. Therefore, the only property which will affect
the pressure is density, which will change by the total propellant mass divided by the tank
volume. Using a total Xenon mass of 150 kg, we come up with a tank pressure drop of
2100 psi. So, to keep the Xenon flowing through mission lifetime, we need an initial tank
charge of 2198 psia.
Thermal Control
We assume that after the tank is charged with Xenon to its initial pressure, it will remain
at room temperature (298 K) until approximately the time the spacecraft exits the launch
vehicle’s payload faring. At this point, the tank will suddenly be exposed to zero
temperature. The role of a heating source for the tank is to maintain the Xenon’s
temperature at room temperature. The heating source will have to offset radiation from
the tank walls out into space. In addition, a certain amount of heat will be lost as a result
of the pressure drop. Equation A-5.3.3-8 below is an expression for radiation, where σ is
the Stefan-Boltzmann constant and ε is the emissivity of the material.
𝑞" = 𝜀𝜎𝑇 4
(A-5.3.3-8)
The emissivity of MLI is assumed to be a conservative 0.01. We obtain a radiation heat
loss by multiplying the expression in Equation A-5.3.3-8 by the surface area of the tank.
This comes out to 2.3 Watts. We calculate the heat lost due to the pressure drop using the
equation of state, and this comes out to 0.9 Watts, giving a total of 3.2 Watts. To be safe,
we allocate 5 Watts for a resistance heating wire.
Author: Brad Appel
Appendix A – 100g Payload – Lunar Transfer
Section A-5.3.3
The thruster will take in about 1400 watts and produces jet thrust with approximately
55% efficiency. In other words, 45% of the input power (630 Watts) is contributing to
some form of energy that doesn’t produce thrust. Two power sinks are considered: the
thermal energy rate drawn for raising the temperature of the Xenon, and heat soak out of
the thruster. The heat lost to Xenon is calculated to be on the order of ten Watts, which
means that just about all of the thrust inefficiency is going to waste heat. See Section A5.3.6 for details on radiator sizing.
Author: Brad Appel