A.5.2.3 Nose Cone1
A.5.2.3.3 Thermal Analysis
Thermal analysis for the nose cone during ascent proves to be the limiting factor throughout the design phase. An initial analysis of the power-law body as originally defined immediately proves that the heating rate at the tip of the nose approaches infinity.This result implies infinite heat transfer to the nose cone throughout flight. As an infinite heating rate is clearly unacceptable, the first step in refinement requires blunting the tip of the nose cone in order to bring the radius of curvature up. The heating rate of a leading body is dependent upon both the physical shape of the object as well as the material properties. Heating rate is primarily dependent upon the radius of curvature of the test body at a specific point as well as the specific heat of the material used. The heating rate of a leading edge body can be theoretically determined using Eq.(A.5.2.3.3.1).1
/ (A.5.2.3.3.1)whereqis the heating rate per unit area (W/cm2), ρ is the density of the fluid (kg/m3), rn is the radius of curvature of test body (m), V is the instantaneous velocity (m/s), cpwis the specific heat of surface material (J/kg-K), and Twis the instantaneous temperature at the surface (K).
We can see from Eq. (A.5.2.3.3.1) that the heating rate is dependent upon trajectory, material and structural parameters. Since our design process does not entail changing the optimal trajectory and therefore the velocity at any point in the launch, we are forced to focus on changes to both the material and structural properties. Ideal design for meeting the thermal requirements entail increasing the radius of curvature throughout the nose cone, especially at the stagnation point, as well as employing a material with a higher specific heat. Eq.(A.5.2.3.3.1) clearly shows that as the radius of curvature at a point decreases, it increases the instantaneous heat transfer, which accumulates throughout the flight. Qualitative analysis alone is able to prove that the original power-law body is unsuited to withstanding high velocity flight, which requires using a simplified thermal analysis model with a blunted tip.
The initial heating rate equation requires a complicated iterative process as well as converting the given heating rate from Eq. (A.5.2.3.3.1) to a heating rate per volume and then an overall temperature. Initial steps to determine this heating rate requires a calculation of both the local atmospheric enthalpy as well as the velocity contribution. The local, atmospheric enthalpy is calculated using Eq. (A.5.2.3.3.2).
/ (A.5.2.3.3.2)where ha is the local, atmospheric enthalpy (kJ/kg), Cp is the specific heat of air, defined as 1003.5 kJ/kg-K and T is the temperature at the desired altitude calculated using Standard Atmosphere tables (K).
The velocity contribution is the 0.5V2 term of Eq. A.5.2.3.3.1, which contributes more to the conditions on the surface of the nose cone due to our high velocity through high altitude/low-density atmosphere. Figure A.5.2.3.3.1 shows the plot of the individual enthalpy terms, as well as their combined value during the launch vehicle’s trajectory. This allows us to determine the local conditions that will have an effect on the heating rate of the nose cone. Figure A.5.2.3.3.1 shows that since we are launching from a balloon at approximately 30km, the local atmospheric enthalpy contributes very little to the overall enthalpy. As expected with a squared term, the velocity contribution increases slowly at first and then rapidly as the velocity continues to increase throughout ascent. While the velocity continues to increase until we reach the desired velocity for our orbit, we only plotted our data through 65 km above Earth. At this altitude the density of the air would be low enough that the air no longer operates under normal heating laws, providing an upper limit for our calculations.
Fig. A.5.2.3.3.1: Enthalpy vs. time for proposed trajectory
(Vincent Teixeira)
The above research and analysis provides important insight into the factors that affect the heating rate and overall temperature gain of the nose cone that we expect during ascent. However, we are ultimately unable to both iterate and integrate the given function to provide an actual temperature vs. time curve for ascent using various metallic alloys. Combining research from Prof. Schneider1 and the tested components of the Vanguard rocket2, we decided to alter the tip of the nose cone for a more favorable thermal survivability. Prof. Schneider simplifies the heating rate calculation by assuming a blunt nosetip that serves as a massive heatsink. Combining this with the Vanguard nose cone design, which used a solid titanium tip, we arrive at the current design, which takes the original power-law body and replaces the sharp tip with a solid blunt tip as shown earlier in Fig.A.5.2.3.1.2.
Ideally, the thickness of the nose cone skin would be determined by a similar thermal analysis in order to provide the minimal mass necessary to protect both the interior of the nose cone and the structural integrity of the nose cone itself. However, since we are unable to compute complete solutions to the thermal loading of the body, we are unable to determine the minimum thickness that our nose cone would need. Instead, we incorporate historical data from the Vanguard rocket3 to define our thickness. We set the outer walls of our nose cone to be 1.75mm thick, which is actually thicker than the 1.651mm (0.065in) nose cone used by the Vanguard rocket.2 As our thermal analysis shows, due to the high-altitude/low-density atmosphere of our ascent, our thermal loading is expected to be less than that of the Vanguard rocket, which was ground-launched.
A.5.2.3.4 Structural Analysis
Once the nose cone is capable of handling the expected thermal loading, we begin to analyze the structural capabilities of the nose. Of primary concern in this analysis is the stagnation pressure on the blunt nose during ascent. Similar to the method used to determine the total enthalpy during the ascent, local atmospheric pressure is calculated as a function of time during the ascent using the Standard Atmospheric Tableswhile dynamic pressure is calculated using the absolute velocity data provided by the Trajectory group. Stagnation pressure is therefore calculated using Eq. (A.5.2.3.4.1).
/ (A.5.2.3.4.1)where P0 is the desired stagnation pressure (Pa), Ps is the local atmospheric pressure from the Standard Atmosphere tables (Pa), ρ is the density of air at the current altitude (kg/m3) and V is the absolute velocity of the launch vehicle (m/s).
Similar to the data gathered for enthalpy during ascent, the local atmospheric pressure contribution is significantly smaller than that of the dynamic pressure, due mostly to the high-altitude launch. Figure A.5.2.3.4.1 plots the stagnation pressure versus time for the launch vehicle during ascent for the 5kg payload. As expected, the local atmospheric pressure drops off quickly as the launch vehicle accelerates through the atmosphere. However, the dynamic pressure curve initially starts at zero and increases quickly as a result of the rapidly accelerating launch vehicle. Since the velocity term is squared, we expect the dynamic pressure to increase rapidly and provide more of a contribution to the stagnation pressure before dropping off as a result of the low-density atmosphere. Combining both values into a maximum stagnation pressure allows us to determine the maximum axial loading for the nose cone.
Fig. A.5.2.3.4.1: Pressure vs. time for proposed trajectory
(Vincent Teixeira)
We initially assume that the solid titanium tip would be structurally capable of supporting the stagnation pressure, which led to determining the need for axial strengthening throughout the lower portion of the nose cone. In order to determine the compressive loading on any stringers placed in the nose cone, we add the maximum expected stagnation pressure to the mass of the solid titanium tip, at which point our reserve factor of safety of 1.25 is taken into account. Initial tests assign the stringers to be made from aluminum in an effort to both reduce cost and mass. In order to write a code that determines the necessary number of stringers to withstand the axial loading, we arbitrarily set the stringer area. For this we choose to use stringers 3mm wide by 10mm deep, similar to those designed throughout the inter-stage skirts of the launch vehicle. Using Eq. (A.5.2.3.4.2), we are able to calculate the required number of stringers to support both the structural mass of the titanium tip as well as the stagnation pressure during ascent, assuming that the titanium/aluminum wall does not carry any axial loading.
/ (A.5.2.3.4.2)where is the yield stress of the stringers (Aluminum 7075), defined as 461 MPa for all calculations, is the assumed maximum G-loading during the flight, which we assume to be 6, is the mass of the solid titanium tip (kg), is the maximum stagnation pressure calculated earlier (Pa), is the area of the blunt tip (m2), is the required number of stringers, and is the area of each individual stringer, arbitrarily set at 30mm2.
Using the 5kg payload as our test case, we find that the nose cone only requires 1.20 stringers to support the required forces. For this calculation, we assume a maximum G-loading of 6, concurrent with that provided by the Trajectory group and used throughout the structural analysis of the entire launch vehicle. Since we clearly cannot have a fraction of a stringer, we decide to include eight stringers in the nose cone, spaced evenly around the circumference in order to support the necessary loading and meet the required factor of safety, set at 1.25 for structural components. This stringer placement remains constant throughout all three launch vehicles in order to provide added axial integrity to the nose cone.
Once the nose cone is capable of withstanding the expected thermal and structural loading, we are able to finally calculate the required mass for the nose cone for each launch vehicle. Table A.5.2.3.4.1 contains the mass of each nose cone.
Table A.5.2.3.4.1: Nose cone massesLaunch Vehicle / Mass of Nose Cone (kg)
200g / 1.7507
1kg / 2.0435
5kg / 1.7927
A.5.2.3.5 Math Model Flowchart
Fig. A.5.2.3.5.1:Nose Cone Math Model Flowchart
(Vincent Teixeira)
References
1Schneider, S.P., “Methods for Analysis of Preliminary Spacecraft Designs.” AAE 450 Spacecraft Design, Sept 2005.
2Klamans, B., “The Vanguard Satellite Launching Vehicle,” The Martin Company, Engineering Report No. 11022, April 1960.
Author: Vincent J. Teixeira