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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:

Image Added

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:

Mathinline
body\text{Heat Flux into surface} = \text{Heat Flux leaving surface}

Mathinline
body\text{Convection In} + \text{Radiation In} = \text{Conduction into material}

Mathinline
body\dot{q}_{conv} + \epsilon \sigma (T_{free}^4 - T_w^4) = \kappa (\frac{dT}{dx})_w

 

where 

Mathinline
bodyT_{free}
 is the free stream air temperature at the altitude of interest, 
Mathinline
bodyT_w
 is the wall temperature, 
Mathinline
body\kappa = \frac{k}{\rho_0 C}
 is the thermal diffusivity which uses 
Mathinline
bodyk
the thermal conductivity, 
Mathinline
body\rho_0
 the material density, and 
Mathinline
bodyC
 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.>

 

https://doi-org.ezproxyberklee.flo.org/10.2514/3.45483

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

Mathinline
body\dot{q}_{conv} = C \rho^N V^M

where 

Mathinline
body\dot{q}_{conv}
 is the heat flux in W/cm^2, 

Mathinline
bodyC = (1.83\times 10^{-8}) \frac{1}{\sqrt{r_n} }(1-g_w)

is a constant that depends on the nose cone tip radius (in meters), and  

Mathinline
bodyg_w
 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), 
Mathinline
body\rho
 is the free stream air density (in kg/m^3) at the flight altitude, and 
Mathinline
bodyV
 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:

Image Added