Rocket Sizing - The Alna Space...

Rocket SizingAlna Space Program

G. Flanagan, Dec. 7, 2011

Introduction

ApproachThese functions perform a first-order approximate sizing for a rocket based on inputs for the required mission change in velocity, the payloadmass, the propulsion system specific impulse, and a series of mass ratios. The overall mass ratio (empty to full mass of a rocket stage) is givenby the basic rocket equation

me

mf= exp -

DV

g ISP

where ISP is the specific impulse, and g is the acceleration due to gravity. me is the empty mass, and mf is the full mass. Note that thisapproach does not account for any losses due to aerodynamic drag, or gravity losses.

In order to break down the system masses further, a mass ratio is specified for the propellant tanks (empty mass of tanks divided by fullweight), and the thrust-to-weight ratio of the propulsion system. We have also added a recovery system mass ratio (mass of recovery system,such as wings, thermal tiles, or a parachute, divided by the total empty mass of the rocket). Finally, a fixed system mass may be included. Thiswould be the mass of equipment that does not scale with the overall mass of the stage. Avionics might be an example of a fixed system mass.These ratios define the mass of these major systems, but an iterative process must be used to find a final solution. For example, the tankweight is specified by the propellant weight, but the propellant weight is a function of the total weight of the system, including the tanks.

The function multiStage is used to determine the mass distribution for a multi-stage rocket. In this case, the payload of for a lower stage is thetotal mass of all the stages above it. multiStage assumes that the user knows the staging velocity. A final function, optimizeStagingVelocity,can be used to determine the staging velocities that will result in a minimum mass system.

DefinitionsThe functions share the following arguments.

payload - the useful mass inserted into orbit.deltaV - the change in velocity for a stage.IPS - specific impulse of the rocket given in Sec. Roughly equals the exhaust velocity divided by the acceleration due to gravity (G). Note thatG is embedded in the functions as m/sec. Therefore, all the inputs should be in metric.tankMassRatio - Empty mass of propellant tanks divided by the tank full mass.thrustToWeight - Thrust-to-weight ratio of the rocket engine divided by the minimum acceleration in G’s. recoverySystemMassRatio - weight of any recovery system (wings, parachutes, thermal system) divided by the total empty weight of thevehicle.

Examples and Discussion

Load package

<< spaceLaunch`rocketSizing`

Show all the functions

? spaceLaunch`rocketSizing`*

spaceLaunch`rocketSizing`

findComponentMasses optimizeStagingVelocity tabulateResult

multiStage printResult

Single Stage caseThe appropriate function for a single stage rocket is findComponentMasses. mutliStage will also work and gives the same results.

? findComponentMasses

findComponentMasses@payload,deltaV,ISP,tankMassRatio,thrustToWeight,recoverySystemMassRatio,systems,optDiterates to find the component masses of a stage, given the payload, deltaV of the state, propulsion ISP,the tankMassRatio

Hempty mass of tank over full massL, thrustToWeight

of propulsion system. recoverySystemMassRatio Hrecovery system mass

over total empty massL and a value for the fixed weight of any systems not accounted for in these ratios. Returns the list

8propellant mass, engine mass, recovery system mass, tank mass<. The possible options is printResult®False�True

Consider a rocket that will accelerate a 1000 kg payload to 8000 m/sec (low earth orbital velocity). The rocket engines have a fairly efficientISP of 350 sec, and a thrust-to-weight ratio of 30. A liquid fuel rocket engine may have a thrust-to-weight of over 60, but we have to divide bya minimum acceleration of 2 g’s. The tank mass ratio (full mass divided by empty mass) is 0.05. There is a recovery system that weights 0.1times the empty mass of the rocket. The systems mass is set to zero.

findComponentMasses@1000, 8000, 350, .05, 30, .1, 0, printResult ® TrueD;

Payload 1000

Propellant Mass 69 267.4

Engine Mass 2557.78

Recovery System Mass 746.595

Prop. Tank Mass 3463.37

Total Mass 76 035.1

Figure 1. Mass breakdown for 1000 kg payload, DV = 8000 m/sec, ISP=350, tank mass ratio =0.05, thrust-to-weight=30, recovery system mass ratio=0.1

Thus, the lift-off mass is 76 times the payload mass. This large ratio is a major part of why rockets for space flight are expensive. Perhapsmost important there are 6000 kg of expensive aerospace structure and equipment for 1000 kg of payload. A modern airliner costs about$350/lb of empty mass (=770/kg) [Collopy and Eames]. That gives a purchase cost of $4620 per kilogram of payload. Presumably, rocketscost a bit more than airliners.

For the specific impulse of 350 sec, a final velocity of 8000 is getting close to the maximum before the launch mass begins to climb exces-sively, as shown by the plot below. The plot uses all the same input parameters with DV as an independent variable. The real DV need toreach orbit is closer to 10,000 m/sec due to aerodynamic and gravity losses.

2 rocket_designer.cdf

http://www.dfmconsulting.com/cost.pdf

Plot@Total�findComponentMasses@1000, v, 350, .05, 30, .1, 0D, 8v, 1000, 12 000<,

PlotRange ® 8All, 80, 5 * 10^5<<, AxesLabel ® 8"DV", "Total Launch Mass, kg"<D

2000 4000 6000 8000DV

100 000

200 000

300 000

400 000

500 000

Total Launch Mass, kg

Figure 2. Lift-off mass for 1000 kg payload, ISP=350, tank mass ratio =0.05, thrust-to-weight=30, recovery system mass ratio=0.1

Besides the rocket engine ISP (approximately the rocket engine exhaust velocity divided by the acceleration due to gravity, giving units ofsec), the lift-off mass is also sensitive to the mass ratio for the propellant tanks. For the popular Delta II stage I booster, the mass fraction is0.06. Recoverable, winged vehicles have higher mass fractions because the propellant tank shapes compatible with aerodynamic shapes maynot be the most structurally efficient. The following plot shows lift-off mass for a series of propellant tank mass ratios.

Plot@Table@Total�findComponentMasses@1000, v, 350, r, 30, .1, 0D,

8r, 8.01, .05, .1, .2<<D, 8v, 1000, 12 000<, PlotRange ® 8All, 80, 5 * 10^5<<,

AxesLabel ® 8"DV", "Total Launch Mass, kg"<D

0.100.20 0.05 0.01

2000 4000 6000 8000 10 000DV

100 000

200 000

300 000

400 000

500 000

Total Launch Mass, kg

Figure 3. Lift-off mass for 1000 kg payload, ISP=350, tank mass ratio =0.05, thrust- to - weight = 30, recovery system mass ratio=0.1 for a series of

tank mass ratios.

Multistage with specified staging velocity The standard method for reducing launch mass is to use stage. First, compute the lift-off mass for a single stage rocket that reaches 10000m/sec, with an ISP of 350 and propellant tank mass ratio of 0.1. The resulting mass is absurdly large.

rocket_designer.cdf 3

findComponentMasses@1000, 10 000, 350, .1, 30, 0, 0, printResult -> TrueD;

Payload 1000

Propellant Mass 1.61702 ´ 108

Engine Mass 5.69978 ´ 106

Recovery System Mass 0.

Prop. Tank Mass 1.61702 ´ 107

Total Mass 1.83571 ´ 108

Figure 4.Mass breakdown for single-stage rocket with 1000 kg payload, DV=10000 m/sec, ISP=350, tank mass ratio =0.05, thrust-to-weight=30.

Now, consider a two stage rocket. The stages are identical, and the staging velocity is 1/2 of the maximum velocity. The resulting lift-off massis much more reasonable.

twoStageV = 85000, 10 000<;

data2 = 88350, .1, 30, 0, 0<, 8350, .1, 30, 0, 0<<;

multiStage@1000, twoStageV, data2, printResult ® TrueD;

Stage 1 Stage 2

Propellant Mass 50 598. 6229.13

Engine Mass 2199.28 270.753

Recovery System Mass 0. 0.

Prop. Tank Mass 5059.8 622.913

Total Stage Mass 57 857.1 7122.8

Total Rocket Mass 65 979.9

Figure 5.Mass breakdown for two-stage rocket with 1000 kg payload, DV=10000 m/sec, ISP=350, tank mass ratio =0.05, thrust-to-weight=30.

We can keep adding stages with identical input parameters, and equally divided the velocity between stages. The total mass continues toimprove, the marginal gain is small after four stages.

TableForm@Table@

8n,

Total�Flatten�multiStage@1000, Table@10 000 * i � n, 8i, 1, n<D,

Table@8350, .1, 30, 0, 0<, 8n<DD<,

8n, 1, 5<D, TableHeadings ® 8None, 8"Stages", "Total Mass"<<D

Stages Total Mass

1 1.83571 ´ 108

2 64 979.9

3 42 974.3

4 38 948.2

5 37 831.7

Figure 5.Total mass for multi-stages rockets with 1000 kg payload, DV=10000 m/sec, ISP=350, tank mass ratio =0.1, thrust-to-weight=30.

For a more efficient tank mass ratio of 0.05, there is less advantage to throwing away empty tanks, and the improvements flatten out afterthree stages.


TableForm@Table@

8n,

Total�Flatten�multiStage@1000, Table@10 000 * i � n, 8i, 1, n<D,

Table@8350, .05, 30, 0, 0<, 8n<DD<,

8n, 1, 5<D, TableHeadings ® 8None, 8"Stages", "Total Mass"<<D

Stages Total Mass

1 2.98592 ´ 106

2 37 371.8

3 31 185.2

4 30 051.4

5 30 027.9

Figure 6.Total mass for multi-stages rockets with 1000 kg payload, DV=10000 m/sec, ISP=350, tank mass ratio =0.05, thrust-to-weight=30.

The relationship between the advantage of staging and the structural efficiency is shown in the following plot.

Needs@"PlotLegends`"D;

Plot@8Total�findComponentMasses@1000, 9000, 300, x, 30, 0, 0D,

Total�Flatten�multiStage@1000, Table@9000. * i � 2, 8i, 1, 2<D,

Evaluate�Table@8300, x, 30, 0, 0<, 82<DD,


Evaluate�Table@8300, x, 30, 0, 0<, 83<DD,


Evaluate�Table@8300, x, 30, 0, 0<, 84<DD<, 8x, .001, .15<,

PlotRange ® 8All, 80, 100 000<<,

AxesLabel ® 8"Tank Mass Ratio", "Lift-Off Mass, kg"<,

PlotLegend ® 8"1 stage", "2 stage", "3 stage", "4 stage"<,

LegendPosition ® 8.9, -0.4<, LegendSize ® 8.33, .4<D

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14Tank Mass Ratio

20 000

40 000

60 000

80 000

100 000

Lift-Off Mass, kg

4 stage

3 stage

2 stage

1 stage

Figure 7.Total mass for multi-stages rockets with 1000 kg payload, DV=9000 m/sec, ISP=350, thrust-to-weight=30 as a function of tank mass ratio.

Multistage with optimized staging VelocityThe function optimizeStagingVelocity attempts to minimize the lift-off mass by making treating the staging velocity as a design variable. Ifthe stages have identical input parameters, then the optimum is for each stage to provide the same change in velocity.


optimizeStagingVelocity@1000, 10 000, Table@8350, .1, 30, 0, 0<, 83<D,

printResult ® TrueD;

Stage 1 Stage 2 Stage 3

Staging Vel 3333.33 6666.66 10 000

Propellant Mass 27 318.1 7739.61 2192.74

Engine Mass 1465.81 415.284 117.656

Recovery System Mass 0. 0. 0.

Prop. Tank Mass 2731.81 773.961 219.274

Total Stage Mass 31 515.7 8928.86 2529.67


Figure 8.Mass breakdown for three-stage rocket with 1000 kg payload, DV=10000 m/sec, ISP=350, tank mass ratio =0.1, thrust-to-weight=30 for all stages. Optimized staging velocity

If the stages are not identical, then the staging velocities are not equal. In the following solution, the first stage ISP is 300 sec, second stage is350 sec, and third stage is 400 sec. In this case, there stage DV is larger for a larger ISP.

optimizeStagingVelocity@1000, 10 000,

88300, .1, 30, 0, 0<, 8350, .1, 30, 0, 0<, 8400, .1, 30, 0, 0<<, printResult ® TrueD;


Staging Vel 1737.23 5179.58 10 000

Propellant Mass 18 355.2 12 412.9 3748.25

Engine Mass 1372.35 653.582 176.658


Prop. Tank Mass 1835.52 1241.29 374.825

Total Stage Mass 21 563.1 14 307.7 4299.73


Figure 8.Mass breakdown for three-stage rocket with 1000 kg payload, DV=10000 m/sec, tank mass ratio =0.1, thrust-to-weight=30 for all stages. ISP is 300 sec for stage 1, 350 sec for stage 2, and 400 sec for stage 3. Optimized staging velocity.

Test functions against Saturn 5 data. We can test whether these very simple functions match reality by comparing results against a real-world rocket. Information for the Saturn 5 iseasily found. The Saturn has the advantage of being a conventionally staged rocket. Many current rockets use strap-on solids or partial staging(Atlas family) that make the calculations more complex. Finally, the aerodynamic losses that we’re are not calculating are smaller for a largerocket.

First, we’ll let the functions determine the staging velocity. The data that follows comes from http://www.astronautix.com/lvs/saturnc5.htm.The payload for low-earth orbit is 120,000 kg. The DV to a 185 km orbit is about 8000 m/sec. Aerodynamic and gravity losses are typicallyaround 1500 m/sec, so the total DV will be taken as 9500 m/sec. The stage mass ratios given in the reference are the total empty to full ratios,which includes propulsion and other systems. We’ll use the numbers directly, but there is some double-accounting for the engine mass. Theresult follow.

optimizeStagingVelocity@120 000, 9500,

88304, .059, 33, 0, 0<, 8420, .083, 33, 0, 0<, 8420, .113, 33, 0, 0<<,



Staging Vel 463.384 5409.31 9500

Propellant Mass 369 076. 1.46499 ´ 106 280 714.

Engine Mass 77 715.5 63 516.5 13 513.6


Prop. Tank Mass 21 775.5 121 594. 31 720.6

Total Stage Mass 468 567. 1.6501 ´ 106 325 948.

Total Rocket Mass 2.56461 ´ 106

Figure 9.Mass breakdown for simulation of Saturn 5 with function optimized staging velocities.


http://www.astronautix.com/lvs/saturnc5.htm

The optimization function gives a low velocity for burn-out of the first stage, caused by the relatively low ISP of the stage. Consequently, thefirst stage is much smaller than for the real rocket. The total mass of 2.5 106 kg compares to the actual lift-off mass of 2.85 106 kg.

According to http://en.wikipedia.org/wiki/Saturn_V, the actual first stage velocity is 2300 m/sec. Also, most of the losses occur during thefirst stage burn, so we’ll add 1500 m/sec to 2300 to get 3800 m/sec. We’ll then split the remaining DV between the remaining two stages.

multiStage@120 000, 83800, 3800 + H9500 - 3800L � 2, 9500<,

88304, .059, 50, 0, 0<, 8420, .083, 50, 0, 0<, 8420, .113, 50, 0, 0<<,



Propellant Mass 2.13576 ´ 106 321 454. 141 207.

Engine Mass 59 297.9 12 876.6 5656.4


Prop. Tank Mass 126 010. 26 680.7 15 956.4

Total Stage Mass 2.32107 ´ 106 361 011. 162 820.

Total Rocket Mass 2.9649 ´ 106

Figure 10.Mass breakdown for simulation of Saturn 5 with specified staging velocities.

This approach is closer to reality. The total mass overshoots the actual value by 4%. The first stages mass of 2.3 Mega Kg compares to theactual value of 2.2 Mega Kg. Choosing staging velocity is apparently more complex than minimizing total mass using the first-order rocket-equation approximations. A major part of the problem with our approach is accounting for losses during the early part of the rocket flight.

We can also back-compare the propulsion mass. The F1 engine weight is 18,416 lb.[http://en.wikipedia.org/wiki/F-1_%28rocket_engine%29]. Converting to kg and multiplying by 5 (the number of first-stage engines) gives41854 kg, which compares to 59297 kg from the rough calculation.

Rocket Design WidgetAll the user functions can be rolled into a Manipuate widget.


numberOfStages 1 2 3 4

Log of Payload 3

Final Velocity 8000

optimizeVel

Stage1

ISP 300

Tank Mass Ratio 0.05

Engine Thrust to Wt 50

Recovery System Mass Ratio 0

Log of systems mass 0

Payload 1000

Propellant Mass 332 974.

Engine Mass 7129.95

Recovery System Mass 0.

Prop. Tank Mass 16 648.7

Total Mass 356 752.


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