Rocket Sizing

Alna Space Program

G. Flanagan, Dec. 7, 2011

Introduction

Approach

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

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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 this approach 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 full weight), 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. This would 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 tank weight 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 the total 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.

Definitions

The 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 that G 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 the vehicle.

Examples and Discussion

Load package

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Show all the functions

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Single Stage case

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

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Consider a rocket that will accelerate a 1000 kg payload to 8000 m/sec (low earth orbital velocity). The rocket engines have a fairly efficient ISP 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 by a 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 0.1 times the empty mass of the rocket. The systems mass is set to zero.

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Payload 1000
Propellant Mass 69267.4
Engine Mass 2557.78
Recovery System Mass 746.595
Prop. Tank Mass 3463.37
Total Mass 76035.1

Figure 1. Mass breakdown for 1000 kg payload, ΔV = 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. Perhaps most 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, rockets cost 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 excessively, as shown by the plot below. The plot uses all the same input parameters with ΔV as an independent variable. The real ΔV need to reach orbit is closer to 10,000 m/sec due to aerodynamic and gravity losses.

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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 of sec), 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 is 0.06. Recoverable, winged vehicles have higher mass fractions because the propellant tank shapes compatible with aerodynamic shapes may not be the most structurally efficient. The following plot shows lift-off mass for a series of propellant tank mass ratios.

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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 10000 m/sec, with an ISP of 350 and propellant tank mass ratio of 0.1. The resulting mass is absurdly large.

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Payload 1000
Propellant Mass 1.61702*10^^8
Engine Mass 5.69978*10^^6
Recovery System Mass 0.
Prop. Tank Mass 1.61702*10^^7
Total Mass 1.83571*10^^8

Figure 4.Mass breakdown for single-stage rocket with  1000 kg payload, ΔV=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 mass is much more reasonable.

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Stage 1 Stage 2
Propellant Mass 50598. 6229.13
Engine Mass 2199.28 270.753
Recovery System Mass 0. 0.
Prop. Tank Mass 5059.8 622.913
Total Stage Mass 57857.1 7122.8
Total Rocket Mass 65979.9

Figure 5.Mass breakdown for two-stage rocket with  1000 kg payload, ΔV=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 to improve, the marginal gain is small after four stages.

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Stages Total Mass
1 1.83571*10^^8
2 64979.9
3 42974.3
4 38948.2
5 37831.7

Figure 5.Total mass for multi-stages rockets with 1000 kg payload, ΔV=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 after three stages.

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Stages Total Mass
1 2.98592*10^^6
2 37371.8
3 31185.2
4 30051.4
5 30027.9

Figure 6.Total mass for multi-stages rockets with 1000 kg payload, ΔV=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.

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

Multistage with optimized staging Velocity

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

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Stage 1 Stage 2 Stage 3
Staging Vel 3333.33 6666.66 10000
Propellant Mass 27318.1 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 31515.7 8928.86 2529.67
Total Rocket Mass 43974.3

Figure 8.Mass breakdown for three-stage rocket with  1000 kg payload, ΔV=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 is 350 sec, and third stage is 400 sec. In this case, there stage ΔV is larger for a larger ISP.

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Stage 1 Stage 2 Stage 3
Staging Vel 1737.23 5179.58 10000
Propellant Mass 18355.2 12412.9 3748.25
Engine Mass 1372.35 653.582 176.658
Recovery System Mass 0. 0. 0.
Prop. Tank Mass 1835.52 1241.29 374.825
Total Stage Mass 21563.1 14307.7 4299.73
Total Rocket Mass 41170.6

Figure 8.Mass breakdown for three-stage rocket with  1000 kg payload, ΔV=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 is easily 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 large rocket.

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 ΔV to a 185 km orbit is about 8000 m/sec. Aerodynamic and gravity losses are typically around 1500 m/sec, so the total ΔV 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. The result follow.

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Stage 1 Stage 2 Stage 3
Staging Vel 463.384 5409.31 9500
Propellant Mass 369076. 1.46499*10^^6 280714.
Engine Mass 77715.5 63516.5 13513.6
Recovery System Mass 0. 0. 0.
Prop. Tank Mass 21775.5 121594. 31720.6
Total Stage Mass 468567. 1.6501*10^^6 325948.
Total Rocket Mass 2.56461*10^^6

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

The optimization function gives a low velocity for burn-out of the first stage, caused by the relatively low ISP of the stage. Consequently, the first stage is much smaller than for the real rocket. The total mass of 2.5 rocket_designer_21.gif kg compares to the actual lift-off mass of 2.85 rocket_designer_22.gif 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 the first stage burn, so we’ll add 1500 m/sec to 2300 to get 3800 m/sec. We’ll then split the remaining ΔV between the remaining two stages.

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Stage 1 Stage 2 Stage 3
Propellant Mass 2.13576*10^^6 321454. 141207.
Engine Mass 59297.9 12876.6 5656.4
Recovery System Mass 0. 0. 0.
Prop. Tank Mass 126010. 26680.7 15956.4
Total Stage Mass 2.32107*10^^6 361011. 162820.
Total Rocket Mass 2.9649*10^^6

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 the actual 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) gives 41854 kg, which compares to 59297 kg from the rough calculation.

Rocket Design Widget

All the user functions can be rolled into a Manipuate widget.

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Spikey Created with Wolfram Mathematica 8.0