This article will describe:

1. The thermal considerations in designing a nose cone tip
2. A mathematical model to describe how hot the nose cone tip is expected to get
3. The numerical method used to determine the temperature profile of the tip
4. The final design choice made for Hermes II

Nose cone tip thermal consideration

Hermes II is expected to reach high speeds in low atmosphere: approx Mach 3.5 at just 10,000 ft when burn out occurs. At these high speeds, the stagnation point temperature of the air is extremely high:

The service temperature of aluminium is on the order of 580K, for steel it is 1200K. 

Does that mean that a steel nose cone is sufficient to withstand the flight? Not necessarily. 

We need to understand how the heat flowing into the nose cone tip from the free stream air will heat up the nose cone tip, and ensure that (1) the nose cone tip temperature doesnt exceed the service temperature of the material used and (2) that the temperature of the part connected to the nose cone tip doesnt exceed its service temperature. 

So what is the source of the heat flowing into the tip?

1. The convective heat transfer from the free stream into the tip
2. The radiative heat transfer from the free stream into the tip

And then the heat is conducted into the surface of the tip.

The important energy balance happens at the surface of the tip:

Note, the direction of the arrows may flip based on which direction the heat wants to flow. 

There are three key questions we need to answer for the nose cone tip design:

1. Material of tip
   • max service temperature
   • will it ablate
   • cost
   • weight
   • manufacturability
2. Length of tip
   • longer to ensure the connected parts aren't too hot
   • shorter to reduce mass
3. Tip Radius
   • blunter is better to create a bow shock further from the surface, and therefore reduce the heating at the tip
   • sharper is better to reduce drag (supposedly, but this is generally a weak effect it seems)

Mathematical Model

Surface Energy Balance

We can perform an energy balance at the surface:

where  is the free stream air temperature at the altitude of interest,  is the wall temperature,  is the thermal diffusivity which uses the thermal conductivity,  the material density, and  the specific heat capacity of there material. Therefore, to use this equation to solve for the wall temperature, we would need to know the convective heat transfer rate, and the gradient of temperature at the wall.

Convective Heat Transfer Model

Modelling the convective heat transfer is very very tricky. The most accurate way would be to perform a full unsteady simulation of our rocket flying through the air, and measuring the convective heat transfer rate - a better way is to approximate this using a variety of semi-empirical models that have been developed. We used the paper by Tauber: Aerothermodynamics of transatmospheric vehicles <MICHAEL E. TAUBER, GENE P. MENEES, and HENRY G. ADELMAN.  "Aerothermodynamics of transatmospheric vehicles", Journal of Aircraft, Vol. 24, No. 9 (1987), pp. 594-602.>

This paper suggested that the convective heat transfer into stagnation points of a vehicle (for the Earth atmosphere) can be modelled by:

where  is the heat flux in W/cm^2, 

is a constant that depends on the nose cone tip radius (in meters), and   is the ratio of wall enthalpy to total enthalpy is a correction factor for the presence of the boundary layer (in the worst case its 0, and so we assumed as such for sizing),  is the free stream air density (in kg/m^3) at the flight altitude, and  is the free stream airspeed (in m/s). 

This is a first order model, and is probably not super accurate, but is better than anything else we have at the moment. Its also hard to say whether it over or underestimates the actual heat load, and some form of validation would be very helpful. We also dont have a good way to estimate the heating along the surface of the nose cone, away from the stagnation point.

What is important is the inverse square relationship. Therefore, we can HALVE the heat flux going into the nose cone if we make the tip radius 4 times larger. That might seem bad, but look at the space shuttles - they had a blunt nose for precisely this reason. 

Temperature profile within the tip

While we could do a full 3D unsteady thermal simulation of the tip, we should first develop a first order approximation. I convert the problem into a 1D problem, and apply the convective heat flux we calculated above and radiation. So we model the nose cone as:

and now we can write the heat equation in 1D:

This equation needs boundary conditions:

1. Temperature profile at the beginning of the simulation is same as ambient throughout the material: 
   
2. Temperature gradient at wall is given by the surface energy balance. This condition requires the wall temperature, and thus when implementing it numerically, the temperature at the wall from the previous time step is used:
   
3. No heat flux at the back of the material, and thus we impose the gradient is 0. 
   

if L is the length of the sample. 

We can then solve this for T(x,t) which gives us the full time profile of the temperature in the nose cone tip, and we can check if they meet the minimum engineering requirements. Simple. 

Numerical Method

The scripts to perform the above calculations are now incorporated with the sims team's matlab scripts. The basic script flow is:

1. Grab altitude vs time, and speed vs time data from the simtable (the output of the rocket trajectory simulation)
2. convert altitude into free stream density
3. compute the convective heat flux as a function of time using Tauber's model (be careful of units)
4. numerically solve the heat equation:
   1. initialise the variables
   2. at each time step, 
      1. compute the free stream temperature using the altitude time table
      2. compute the net heat flux into the wall (conv + rad)
      3. compute the temperature profile at the current time step using the previous time steps temperature profile, and apply boundary conditions
   3. Save/graph the results in a human friendly way

Ultimately we get a graph that tells us when and where things get toasty, and we can decide if that is enough for us to take a risk or not.

The numerical scheme used is called FTCS - forward time, centred space. The boundary conditions are Neumann Conditions - ie, the gradient at the space boundaries is imposed. 

A good explanation of how the FTCS scheme works is here. 

1. The initial profile is defined
2. For the next time step:
   1. The interior points are computed using the previous temperature profile
   2. The boundary gradients are used to compute the boundary temperatures

The main problem with FTCS is whether the solver is stable. For the heat equation, a stability analysis can show that the time step () and the space step () must satisfy the following relation:

Therefore, in the numerical scheme I have made, I have defined the spacial resolution we are after, and the software computes a suitably small time step to be used. Making the time step and space step smaller is benefitial in reducing the numerical error, but (because my code is rather inefficiently written) will increase the computation time significantly. Worth trying to check though