We discuss in detail all of the required aerodynamic and aerothermal information that is needed in order to launch a 200g, 1kg, and 5kg payload into low earth




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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 than linearly 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. [http://www.astronautix.com/lvs/vanguard.htm]


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


A.1.2.3 Aerodynamic Coefficients


A.1.2.3.1 Drag Coefficient

The coefficient of drag CD is one of the most important aspects of launch vehicle aerodynamics. This small, non-dimensional number impacts many features of the overall launch vehicle design. A few examples include the amount of thrust needed for an appropriate thrust to weight ratio, the overall ΔV required to reach orbit, and the ability to control the launch vehicle. The CD is essentially a means of representing the impact a launch vehicle’s shape will have on the amount of drag incurred as the launch vehicle speeds through the atmosphere. The manner in which the CD achieves this impact can be seen through Eq. (A.1.2.3.1.1)



(A.1.2.3.1.1)

where D is the total drag, q is the dynamic pressure, and A is the area.


One of the first goals of the aerothermodynamics group was to further understand the impact of launch vehicle geometry, Mach number, and angle of attack on the CD. Doing so would allow us to put some preliminary limits on certain aspects of the launch vehicle design, such as diameter, and maximum tolerable angle of attack.


The CD is highly dependent upon Mach number. In the subsonic regime CD is relatively low. In the transonic regime it raises to its highest value, and in the supersonic regime it reduces back to a lower value. An example of this trend is shown in Fig. A.1.2.3.1.1.


Fig. A.1.2.3.1.1: Impact of Mach number on CD for V2 rocket.1

(Jayme Zott)


Not only is the CD defined by the speed of the launch vehicle, it is also defined by aspects of the geometry such as diameter, number of fins, and length. Referencing data from the Vanguard and other historically successful launch vehicles, we realized that as the rocket diameter increased, so did the CD. To get an idea of exactly how much the CD increased with respect to diameter, we referenced established model rocket programs. Table A.1.2.3.1.1 shows outputs from the Aerolab3 model rocket program using Vanguard geometry with varying diameter.


Table A.1.2.3.1.1 The impact of Diameter on CD







Base Diameter

Units

Max CD

1.00

m

0.37

1.14*

m

0.42

1.25

m

0.47

1.50

1.75

2.00

2.25

2.00

m

m

m

m

m

0.70

1.10

1.50

2.10

2.70










Footnotes: * Vanguard base diameter

(Jayme Zott)


From this information we were able to determine that our final launch vehicle diameter should not exceed 2.00 meters in length. Doing so would lead to undesirable CD values in the transonic regime.


The angle of attack also has a noticeable impact on the CD. In order to deduce the magnitude of this impact, we referenced historical data from various launch vehicles.1,2 Using this historical data, we created general trends for the subsonic, transonic, and supersonic regimes shown in Eqs. (A.1.2.3.1.2), (A.1.2.3.1.3), and (A.1.2.3.1.4) respectively.

(A.1.2.3.1.3)

(A.1.2.3.1.2)





(A.1.2.3.1.4)



where CD is the coefficient of drag, CD0 is the initial coefficient of drag, and M is the Mach number, and α is the angle of attack.


By extrapolating these empirical results we were able to show the impact of a wide variety of angles of attack on CD.



Fig. A.1.2.3.1.2: Impact of Angle of Attack on CD.3

(Jayme Zott)


Knowing historical trends for the impact of Mach number, launch vehicle geometry, and angle of attack on the CD is of great use in preliminary analysis. When the team began work on creating a final design configuration, it was necessary to solve for the CD in a much more refined manner. In order to take into consideration all elements of the launch vehicle geometry, angle of attack, and Mach number for the final design analysis, linear perturbation theory was used.


Linear perturbation theory is the method in which the pressure over the top and bottom surfaces of the launch vehicle is integrated to solve for axial and normal force coefficients acting on the launch vehicle. From these axial and normal force coefficients, we are then able to use Eq. (A.1.2.3.1.5) to solve for the CD.

(A.1.2.3.1.5)




where CD is the coefficient of drag, CN is the normal force coefficient, CA is the axial force coefficient, and α is the angle of attack. An explanation of how linear perturbation theory is implemented can be found in the following sections on aerodynamic forces, A.1.2.3.2-A.1.2.3.7.

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