Page History

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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 

LaTeX Math Inline
body T_{free}

 is the free stream air temperature at the altitude of interest, 

LaTeX Math Inline
body T_w

 is the wall temperature, 

LaTeX Math Inline
body \kappa = \frac{k}{\rho_0 C}

 is the thermal diffusivity which uses 

LaTeX Math Inline
body k

the thermal conductivity, 

LaTeX Math Inline
body \rho_0

 the material density, and 

LaTeX Math Inline
body C

 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.>

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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:

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LaTeX Math Inline
body \frac{dT}{dt} = \kappa \nabla^2 T = \kappa \frac{d^2 T}{dx^2}

This equation needs boundary conditions:

1. Temperature profile at the beginning of the simulation is same as ambient throughout the material: 
   

   LaTeX Math Inline
   body T(x,0) = T_{amb}

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:
   

   LaTeX Math Inline
   body \frac{dT}{dx} ( 0,t) = \frac{1}{\kappa} \left( \dot{q}_{conv} + \epsilon \sigma (T_{free}^4 - T(0,t)^4)\right)

3. No heat flux at the back of the material, and thus we impose the gradient is 0. 
   

   LaTeX Math Inline
   body \frac{dT}{dx}(L,t) = 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 (

LaTeX Math Inline
body \Delta t

) and the space step (

LaTeX Math Inline
body \Delta x

) must satisfy the following relation:

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body r = \frac{\kappa \Delta t}{\Delta x^2} < \frac{1}{2}

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 

 

 

 

  is solved the gradient $dT/dx$ can be found