A.5.2.3 Nose ConeA.5.2.3.1 OverviewThe final nose cone design revolves around a power-law body, with a blunted tip in order to reduce the effects of heating throughout ascent. During original rocket design, drag losses provided a severe limitation to the capabilities and mission parameters. Power-law bodies are the optimum shape for minimum drag when it comes to the leading edge of a body of revolution.1 A power-law body is a body of revolution whose revolving surface is governed by Eq. (1) below.

where: x is the position measured along axis of symmetry, L is the total axial length of the power-law body, R is the radius of the body at the end-point, m is a pre-determined power-law body coefficient, and r is the radius of power-law body at axial position xTo start with, we chose a power-law coefficient (m = 0.7) that corresponded with the lowest drag achieved during Auman and Wilks experiments.1 Further design steps required defining the length of the nose cone as well as thermal and structural analyses. The initial length of the nose cone was set to 1.5 times the radius of the base. Using this as a starting point, we attempted to reach a balance between elongating the nose cone to reduce drag and shortening the length in order to reduce the nose cone heating. Since the nose cone will be subjected to a high heating rate due to the velocity through the atmosphere, we decided that further elongating the nose cone would increase overall cost due to the heightened thermal requirements. As Eq. (1) shows when x approaches zero, which is defined as the tip of the nose cone, the radius also approaches zero, resulting in an increasingly sharp tip. Figure 1 shows the general, two-dimensional outline of a power-law body using m = 0.7.

Figure 1 : Outline of Power-law BodyFigure 1 shows that the tip of the initial design reaches a relatively sharp tip. Initial thermal analysis showed that this would be unacceptable, leading to a change in the shape near the tip. Final nose cone design calls for a solid-blunted tip set approximated 1/3 of the way back from the tip of the nose cone. Figure 2 shows the CATIA model of the final nose cone.

Figure 2 : Final Nose Cone DesignSelecting materials capable of handling the thermal loading was the final step in the design of the nose cone. Since titanium has such a high melting temperature, especially in relation to other metallic alloys currently available, it was the necessary choice for the upper half of the power-law body and blunted tip of the cone, which will be subjected to the greatest thermal loading. Since we can assume that the thermal loading lessens as we move further from the tip (as discussed in A.5.2.3.3), it is possible to use aluminum for the lower third of the nose cone. Final considerations for the nose cone involved necessary internal structures to support both the static and expected dynamic loading during flight. While the nose cone is located at the stagnation point during nominal flight, calculated pressure loadings are low enough to negate the use of excessive internal supports. Four internal aluminum stringers are placed symmetrically around the nose cone in order to support the weight of the blunted titanium tip and are capable of withstanding the expected dynamic loads within the reserve factor of 1.25.Further work into the design of the nose cone should focus on the use of ablatives and current software available for the thermal analysis of bodies subjected to the expected thermal loads. Most contemporary space launches employ ablative shells over leading surfaces like the nose cone in order to reduce the necessity of using expensive, difficult materials such as titanium. Further research into the use of ablatives may open up alternatives for the materials used throughout the nose cone. Unfortunately, we were advised that detailed analysis using Sandia One-Dimensional Direct and Inverse Thermal (SODDIT) would be outside the scope and deadlines of this design phase, which left us limited to metallic alloys.

A.5.2.3.3 Thermal AnalysisThermal analysis for the nose cone during ascent proved the limiting factor throughout the design phase. An initial analysis of the power-law body as originally defined immediately proved that the heating rate at the tip of the nose would approach infinity, implying infinite heat transfer to the nose cone throughout flight. As an infinite heating rate was clearly unacceptable, the first step required 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. (3) below.

where q is the heating rate per unit area, is the density of the fluid, rn is the radius of curvature of test body, V is the instantaneous velocity, cpw is the specific heat of surface material, and Tw is the instantaneous temperature at surface.We can see from Eq. (3) that the heating rate is dependent upon trajectory, material and structural parameters. Since our design process did not entail changing the optimal trajectory and therefore the velocity at any point in the launch, we were forced to focus on changes to both the material and structural properties. Ideal design for meeting the thermal requirements would 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 (3) 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 was able to prove that the original power-law body was unsuited to withstanding high velocity flight, which required 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. (3) to a heating rate per volume and then an overall temperature. Initial steps to determine this heating rate required a calculation of both the local atmospheric enthalpy as well as the velocity contribution. The local, atmospheric enthalpy is calculated using Eq. (4) below:

where ha is the local, atmospheric enthalpy, 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.

The velocity contribution is the 0.5V2 term, 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 3 shows the plot of the individual enthalpy terms as well as theyre combined value. This allows us to determine the local conditions that will have an effect on the heating rate of the nose cone. Figure 3 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.While the current research provided important insight into the factors that affect the heating of the nose cone throughout ascent, we were 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. Schneider2 and the tested components of the Vanguard rocket3, 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 arrived 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 Figure 2.A.5.2.3.4 Structural AnalysisOnce the nose was capable of handling the thermal loading expected during ascent, we began to analyze the structural properties of the nose cone and how well it would survive the physical loading due to ascent. Of primary concern in this analysis was 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 was calculated as a function of time during the ascent using the Standard Atmospheric Tables while dynamic pressure was calculated using the absolute velocity data provided by the Trajectory group. Stagnation pressure was therefore calculated using Eq. (5) below:

Where Ps is the desired stagnation pressure, Pa is the local atmospheric pressure from the Standard Atmosphere tables, is the density of air at the current altitude and V is the absolute velocity of the launch vehicle.

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 5 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 allowed us to determine the maximum axial loading for the nose cone.We initially assumed 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 rest of the nose cone. In order to determine the compressive loading experienced by any stringers placed in the nose cone, we added the maximum expected stagnation pressure to the mass of the solid titanium tip, at which point our factor of safety was taken into account. Initial tests assigned the stringers to be made from aluminum in an effort to both save money and mass. In order to write a code that determined the necessary number of stringers to withstand the axial loading, we had to arbitrarily set the stringer area. For this we chose to use stringers 3mm wide by 10mm deep, similar to those used throughout the interstage skirts of the launch vehicle. Using Eq. (6) below, we were 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.

Using the 5kg payload as our test case, we found that the nose cone only required 1.20 stringers to support the required forces. Since we clearly cannot have a fraction of a stringer, we decided to include four stringers in the nose cone, spaced evenly around the circumference in order to support the necessary loading and provide a reasonable factor of safety.Once the nose cone was capable of withstanding the expected thermal and structural loading, we were able to finally calculate the required mass for the nose cone for each launch vehicle. Table 1 contains the mass of each nose cone.Table 1: Nose Cone Masses

Launch VehicleMass of Nose Cone (kg)

200g1.7507

1kg2.0435

5kg1.7927

Author: Vincent J. Teixeira