A.1.2.2 Initial Sizing Function 1

A.1.2.2 Initial Sizing Function 1

The purpose of this part of the project is to come up with a method for determining the shape of the launch vehicle. The first method we use is to linearly scale the vehicle by payload mass. To accomplish this, we use the dimensions of two rockets for data points to make the sizing functions; the Vanguard rocket and the Purdue Hybrid Launch Vehicle. This method, however, is ineffective at sizing the vehicle because it yields unrealistic overall lengths for small payload masses. We choose to abandon the linear scaling in favor of sizing the vehicle based on the volume of propellant in each stage.

The method of sizing the vehicle based on fuel volumes yielded realistic lengths for every vehicle. The size was more realistic because it was based off of how much propellant each stage needs instead of a scaling factor based off of the payload mass. However, we had to manually optimize the length and diameter of the vehicle to obtain the final vehicle dimensions. Since this proved time consuming, we discontinued use of the Excel version due to a similar method employed in a large optimization code (MAT code).

To begin the initial sizing of the vehicle, a sizing function was needed. We decided to size the vehicle by linearly scaling the Vanguard rocket based on payload mass. The linear relationship was calculated using Vanguard payload mass data along with stage length and diameter data found from an online source for historical rockets.1 For a second set of data points, the payload mass, stage length, and stage diameter data from the Purdue Hybrid Launch Vehicle were used.2 This data was then entered into Excel and a linear relation between length and diameter was found with respect to payload mass for each stage.An example of how the sizing functions were calculated is shown in Fig. A.1.2.2.1 below.

Figure A.1.2.2.1: Sizing function regression plot for vehicle second stage.

(Chris Strauss)

Figure A.1.2.2.1 shows the regression plot for the length of the second stage of the launch vehicle along with the sizing function associated with the stage. This was created by entering the data for second stage length of Vanguard and the Purdue Hybrid Launch Vehicle versus the payload mass of each. A linear regression line between the points was then plotted and the equation of the line was used as the sizing function for the stage length where x is the payload mass.

We used a similar method on each stage length and diameter until a complete set of dimensions was calculated for each launch vehicle, and each different payload mass. The results of this scaling are shown in Table A.1.2.2.1below for the overall length of the rocket. The results by stage are shown in Tables A.1.2.2.2 through A.1.2.2.4.

Table A.1.2.2.1 Initial Scaling Overall Length vs. Payload Mass
Variable / Value / Units
Payload Mass 1 / 0.20 / kg
Payload Mass 2 / 1.00 / kg
Payload Mass 3 / 5.00 / kg
Overall Length 1 / 0.51 / m
Overall Length 2 / 2.56 / m
Overall Length 3 / 12.78 / m
Footnotes: 1,2,3 for lengths refer to masses 1,2,3
Table A.1.2.2.2Initial Scaling Stage 1 Dimensions vs. Payload Mass
Payload (kg) / Length (m) / Diameter (m)
0.20 / 0.25 / 0.70
1.00 / 1.26 / 0.74
5.00 / 6.32 / 0.94
Footnotes:
Table A.1.2.2.3Initial Scaling Stage 2 Dimensions vs. Payload Mass
Payload (kg) / Length (m) / Diameter (m)
0.20 / 0.11 / 0.39
1.00 / 0.57 / 0.43
5.00 / 2.85 / 0.63
Footnotes:
Table A.1.2.2.4Initial Scaling Stage 3 Dimensions vs. Payload Mass
Payload (kg) / Length (m) / Diameter (m)
0.20 / 0.06 / 0.12
1.00 / 0.28 / 0.15
5.00 / 1.38 / 0.33
Footnotes:

From the data presented, we found that a linear scaling method was not a good method to use. As evidence for this, we looked at the overall length for the 200 gram payload mass and found that it was unreasonably small at 0.51 meters. The only reasonable dimensions calculated using this method were those for the 5 kilogram payload where the overall length was nearly half that of the Vanguard rocket. This is reasonable for a launch vehicle size considering the smaller payload that will be carried. A new method for sizing the launch vehicle needed to be devised to provide more accurate results reflecting the actual size of the launch vehicle and payload.

After the linear scaling method was proven to be very inaccurate, we based our next attempt at sizing the vehicle based on fuel volume. This method relied on finding the amount of fuel burned for each stage and the densities of the fuel being burned. This information was provided by the propulsion group.Below are tables showing the results of sizing the vehicle based on fuel volume.

Table A.1.2.2.5 5 kg Payload Launch Vehicle Dimensions for Various Fuel Combinations
Fuels / Length (m) / Diameter (m)
LOX/HTPB / 5.04 (stage 1) / 3.00 (stage 1)
7.63 (stage 2) / 4.00 (stage 2)
1.88 (stage 3) / 1.00 (stage 3)
H2O2/RP-1 / 9.25 (stage 1) / 6.00 (stage 1)
5.57 (stage 2) / 4.00 (stage 2)
2.33 (stage 3) / 0.75 (stage 3)
AP/HTPB/Al / 16.41 (stage 1) / 6.00 (stage 1)
9.36 (stage 2) / 4.00 (stage 2)
2.63 (stage 3) / 0.75 (stage 3)
Footnotes:
Table A.1.2.2.6 1 kg Payload Launch Vehicle Dimensions for Various Fuel Combinations
Fuels / Length (m) / Diameter (m)
LOX/HTPB / 10.35 (stage 1) / 6.00 (stage 1)
6.28 (stage 2) / 4.00 (stage 2)
2.75 (stage 3) / 0.75 (stage 3)
H2O2/RP-1 / 10.96 (stage 1) / 5.00 (stage 1)
4.58 (stage 2) / 4.00 (stage 2)
1.92 (stage 3) / 0.75 (stage 3)
AP/HTPB/Al / 13.51 (stage 1) / 6.00 (stage 1)
4.93 (stage 2) / 5.00 (stage 2)
2.17 (stage 3) / 0.75 (stage 3)
Footnotes:
Table A.1.2.2.7 0.2 kg Payload Launch Vehicle Dimensions for Various Fuel Combinations
Fuels / Length (m) / Diameter (m)
LOX/HTPB / 14.26 (stage 1) / 5.00 (stage 1)
6.01 (stage 2) / 4.00 (stage 2)
2.64 (stage 3) / 0.75 (stage 3)
H2O2/RP-1 / 10.48 (stage 1) / 5.00 (stage 1)
4.38 (stage 2) / 4.00 (stage 2)
1.83 (stage 3) / 0.75 (stage 3)
AP/HTPB/Al / 12.93 (stage 1) / 6.00 (stage 1)
4.72 (stage 2) / 5.00 (stage 2)
2.08 (stage 3) / 0.75 (stage 3)
Footnotes:

From the data shown in Tables A.1.2.2.5 through A.1.2.2.7, it can be seen that the vehicle sizes are all comparable to each other when similar diameters are used. This implies that a single launch vehicle could be used for all three payloads. This conclusion is based on very minimal optimization of each stage diameter, however. This data also shows that the method of sizing the vehicle based on fuel volume provides us with better results thanlinearly scaling the vehicle based on payload mass. Since the vehicle has realistic lengths, this method could be used for a more in depth sizing analysis once a particular fuel combination is chosen for each stage.

This exact method for determining the size of the vehicle’s stages is not used as the final sizing method, however, since an automatic size optimization routine is included into the MAT code. The MAT code is then used for all sizing problems through the end of the design process.

References:

1Wade, M., “Vanguard”, 1997-2007.[

2Tsohas, J., “AAE 450 Spacecraft Design Spring 2008: Guest Lecture Space Launch Vehicle Design”, 2008

Author: Chris Strauss