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Numeric analysis of nose cone heating: first steps

Calvin and Hobbes on engineering design

Calvin and Hobbes on engineering design

The most accurate way to find the aerodynamic heating on a high velocity flight structure is to fly it and measure what happens. This approach can be expensive, though, requiring building more and more robust rockets until finally one isn’t destroyed. Crazy as it may sound, it is an approach that I’ve seen used.

Another approach is to model the rocket in Computational Fluid Dynamics (CFD) software. This works well for steady state processes, but is incredibly processor-intensive for a time-dependent process such as trying to find the total heating of a nose cone in a flight trajectory with continuously changing air velocity, pressure, and temperature. A steady state analysis can be done with high fidelity results in under an hour, but a 40 second flight simulation requires about a month of processor time.

Least accurate, but easiest, is using empirical models. Rockets and planes have been flown supersonic many times, and from that dataset researchers have created simple equations that can crudely predict the temperature and heating that will be seen in flight.

Recovery temperature

In a previous post I showed the stagnation temperature for a set of flight conditions. Stagnation temperature is the highest temperature of the gas possible, converting all of its kinetic energy into heat.

A more accurate baseline number is the “recovery temperature”, which is related to stagnation temperature but reflects the fact that the conversion to heat from dynamic pressure isn’t total. From Tactical Missile Design, by Eugene L. Fleeman:

TRecovery = TFree Stream ( 1 + 0.2 r M2 )

Where:

  • TRecovery is the recovery temperature, in Kelvin or Rankine.
  • TFree Stream is the free stream temperature; for the nose cone it’s the ambient air temperature at the given altitude, in Kelvin or Rankine.
  • r is the recovery factor. For stagnation r = 1; turbulent boundary layer r = 0.9; Laminar boundary layer r = 0.8. We’re in the turbulent regime for our analysis.
  • M is the mach number.

Adding that to the spreadsheet used previously:

Altitude (m) Temp (K) Pressure (Pa) Density (kg/m^3) Dynamic press (Pa) Mach number Stagn. temp (K) Recovery temp (K)
827 282.8 91777 1.131 66898 1.02 341.7 335.8
2022 275.0 79278 1.004 142116 1.60 415.9 401.8
8056 235.8 35312 0.522 33808 1.17 300.3 293.8
9955 223.4 26619 0.415 126247 2.60 526.2 495.9
11750 216.7 20108 0.323 199903 3.77 831.9 770.4
13703 216.7 14771 0.238 247967 4.90 1255.6 1151.7

As you can see, the recovery temperature is a bit lower than the stagnation temperature, but not hugely. It’s still seeing very hot air when it’s flying at Mach 4.9 at 13.7km.

Heat flux

Tactical Missile Design also provides a very imperial empirical equation for calculating the heat flux the nose cone will see.

Q = 345 ρ0.8 M2.8/x0.2

Where:

  • Q is the heat flux in BTU per square foot per second.
  • ρ is the free stream air density, in slug per cubic foot.
  • M is the mach number.
  • x is the distance from the tip, in feet.

With this equation we get the following chart for the heating of the nose at various positions at the high velocity data point:

Calculation of heat flux

400 BTU/ft^2/s is 4.5 megajoules/m^2/s. It’s a lot of heating. Rounding the tip significantly reduces the heating and causes a minimal increase in drag coefficient.

Converting to temperature rise

As before, we need material properties and part dimensions to convert the heating rate into a peak flight temperature of the part. We will tackle that in a subsequent post.

What of Sugar Shot?

Ultimately we’re trying to establish what materials will be needed for this theoretical Sugar Shot rocket and its very high flight speeds low in the atmosphere. Fleeman has this to say about that:

Figure 4.14

Examples of uninsulated structure materials that are cost effective for the short duration flight of tactical missiles are shown in Fig. 4.14. An assumption is that the heat conducted to the airframe is large compared to the heat radiated by the airframe. This assumption is most applicable at low altitude/high atmospheric density. The airframe temperature would be lower at high altitude, due to the relative importance of radiation from the airframe. The example airframe materials selected for the figure are based on the consideration of weight, cost, and maximum temperature capability. Composite materials are a new technology that will find increased use in new missile airframe structure. High temperature composites have particular benefits for hypersonic missiles, providing weight reduction. Titanium alloy technology also enables lighter weight missiles in a hypersonic, high temperature flight environment.

As shown in the figure, at subsonic and low supersonic Mach number, graphite/ epoxy and aluminum or aluminum alloys are attractive choices for a lighter weight structure without external insulation. Graphite/epoxy and aluminum alloys have high strength-to-weight ratio, are easily fabricated, have good corrosion resistance, and are low cost. For higher Mach number, graphite/polyimide composite structure has an advantage of high structure efficiency at higher temperature for Mach numbers to about Mach 4. For flight to about Mach 4.5 without external insulation, titanium structure and its alloys are preferred. A disadvantage of a titanium structure is higher material and machining cost. For example, a titanium part has a material cost that is up to 18 times that of aluminum and a machining cost that is up to 13 times that of aluminum. However, the cost to cast a part made of titanium is comparable to the cost to cast an aluminum part. Small tolerance (e.g., +/−0.001 in.) is required to avoid expensive touchup machining. Up to Mach 5.7 without external insulation (about 2000◦F), super nickel alloys such as Inconel, Rene, Hastelloy, and Haynes must be used. Precision casting should be used to minimize the expensive machining and material cost associated with super alloys. Above Mach 5.7 the super alloys require either external insulation or active cooling. Active cooling is usually not cost effective for tactical missiles.

This suggests that if a rocket is built with the performance expected, the nose cone may need to be a short blunted Inconel tip attached to a high temperature composite nose cone using state-of-the-art materials. Or it could be a lower temperature composite nose cone covered with an ablative material.

Hypersonics are not a project for fiberglass and epoxy.

Addendum to nose cone heating: stagnation temperature

It’s useful to calculate the stagnation temperature of the air. It gives an upper bound to the temperature that the nose could reach.

Altitude (m) Velocity (m/s) Temp (K) Pressure (Pa) Mach number Stagnation temp (K)
827 344 282.8 91777 1.02 341.7
2022 532 275.0 79278 1.60 415.9
8056 360 235.8 35312 1.17 300.3
9955 780 223.4 26619 2.60 526.2
11750 1112 216.7 20108 3.77 831.9
13703 1445 216.7 14771 4.90 1255.6

Going near Mach 5 at 13km: quite toasty.

Solidworks Simulation of near hypersonic nose cone

Richard asked if I could do some simulation of the nose cone heating of the Double Sugar Shot rocket. It’s quite a big nose cone, nearly a meter long by 169mm diameter (~3ft by 6.65″). I modeled it in Solidworks, then ran it in Solidworks Simulation. With my current computer, it takes a few days for it to complete a run, then a few days more to get time to check the results and re-run it if anything was off. Following is the final results of one run.

I was given a few points with altitude, velocity, temperature, pressure, and density. I calculated dynamic pressure.

Altitude (m) Velocity (m/s) Temp (K) Pressure (Pa) Density (kg/m^3) Dynamic pressure (Pa) Mach number
827 344 282.8 91777 1.131 66898 1.02
2022 532 275.0 79278 1.004 142116 1.60
8056 360 235.8 35312 0.522 33808 1.17
9955 780 223.4 26619 0.415 126247 2.60
11750 1112 216.7 20108 0.323 199903 3.77
13703 1445 216.7 14771 0.238 247967 4.90

The last point had the highest dynamic pressure (also known as “max Q”) for the data points given, so I ran it in the simulation as it may have the most interesting results. That point is 1445 m/s at 13703 m MSL. For the simulation input I used air as the fluid at the density at that altitude; 14771 Pa and 216.65 K. I set the surface roughness to 6.35 micrometer, which should be appropriate for a finish equivalent to smooth paint.

Here’s a video of one result, followed by charts of others. Click to embiggen.

The mesh:Image

 

Gas density:Image

 

Pressure:Image

 

Temperature:Image

 

Velocity:Image

 

The shockwave coming off the cone is prominent in all the plots, and raked back at a severe angle due to the high velocity. The temperature is relatively low, aside from a small warm area just downstream of the nose.

Here the analysis wasn’t set to deal with conduction in solids, so it’s impossible to give temperatures for the solid part, just for the gas touching the part. It does bound the maximum temperature the part could reach, which is a reasonable design point, but it isn’t as interesting as having a plot of the actual expected part temperatures.

To analyze the heat going into the nose cone its material properties must be defined. These include:

  • Density
  • Specific heat (possibly at multiple temperatures)
  • Conductivity type (Isotropic, Unidirectional, Asymmetrical/Biaxial, or Orthotropic)
  • Thermal conductivity (which can also vary with temp and can be multiple, if not isotropic)
  • Melting temperature (the glass transition temperature of the resin would be a good analog for a composite nose cone)

The thickness of the nose cone would also need to be known, especially if it varies at different positions.

For the next analysis I will shorten the length of the straight tube to just a few inches, close the end so that the flow inside isn’t taking calculation cycles, and possibly change the size of the bounding box to remove the volume downstream of the nose.

A full simulation can also be done with time dependent variables. The velocity, temperature, and pressure can all be varied with time to get an idea of the total heating over the flight. However that analysis would probably take weeks for the computer to run. It’d be interesting to try, though.

Image

Colorful fluid drawing of a cone traveling hypersonically

Colorful fluid drawing of a cone traveling hypersonically

Running an experiment with some very basic geometry to see the capabilities of Solidworks Flow Simulation in analyzing a shape traveling mach 5+.

The cone is 4 inches tall and 2 inches wide at the base, to roughly simulate the tip of a rocket nose cone at a severe Max-Q. Fluid is air at STP.

2013 Sparkfun Autonomous Vehicle Competition Recon

Paul Breed asked me to check out a couple things about the course for the 2013 Sparkfun AVC, so I went to check it out.

Here’s the ground course video:

The course generally slopes upward from the start line to the third corner, then slopes back down to the fourth corner.

There’s also a gallery of photos on Flickr. Here are a few of them:

IMG_6129.jpg
Location of potholes on the ground course.

Puddle depth action shot
Pothole depth.

Crack in the straight after turn one, there's a series of them
Size of the largest cracks. These will probably close up somewhat as it gets warmer.

Looking at the trees from the dock
Size of the trees over the aerial path. They’re forty or fifty feet tall.

I hope this helps any potential entrants.

Loadcell adapter plates and the challenges of load constraints

Richard Nakka asked that I run FEA on some plates he’d designed to convert a button-type loadcell into something that could be used for 5000lbf tension testing. It consisted of three 0.375 aluminum plates. Two identical ones on the outside that I named “bread”, and one larger one on the inside that I named “meat”.

I wanted to show the difference in FEA between loading the easy way, into faces of the single part, vs. a more accurate way, into a simulated bolt and loadcell button.

First, here is the meat plate loaded with 5000lb stretching it, with the entire inner surface of the drilled hole applying 5000lb to the right and the rectangular surface of the rounded square hole fixed:

Factor of safety:

Tensile stress:

The plots show that the factor of safety is generally greater than 1, so it won’t deform the part. But while this is a very easy simulation to set up, such that you can do it in the Solidworks “SimulationXpress”, it isn’t a realistic simulation. Essentially, what you’re saying here is that when you put a bolt through the hole it will not only push on one side of the hole, but also be bonded to and pull on the other side of the hole. Bolts don’t do that.

To set up a more realistic simulation, I created parts to mock up a loadcell button and a bolt through the aforementioned hole. Here are the same plots, but instead with the force applied to the back of the button and the bolt fixed on either end:

Factor of safety:

Tensile stress:

The scales are the same in both sets of images. The latter set shows that applying the force through a button and 0.24″ rod significantly changes the stresses in the part. The aluminum plate will definitely yield behind the bolt, though it probably won’t fail. This is a result you can only really find with the extended Solidworks Simulation, as Simulation Xpress doesn’t allow multi-part assemblies.

Finally, and less interesting, here are the same plots for the ‘bread’ plates:

Factor of safety:

Stress:

And finally a view of how the part will stretch as a spring, with the colors representing how far the various areas of the part will move from their original location: