Nose Cone Tip Thermal Analysis

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:

$//<![CDATA[ \begin{array}{l}T_{stag} = T \left(1 + \frac{\gamma - 1 }{2} M^2 \right) = 268 \left( 1 + \frac{1.4 - 1 }{2} 3.5^2 \right) = 924 K\end{array} //]]>$

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:

$//<![CDATA[ \begin{array}{l}\text{Heat Flux into surface} = \text{Heat Flux leaving surface}\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}\text{Convection In} + \text{Radiation In} = \text{Conduction into material}\end{array} //]]>$

$//<![CDATA[ \begin{array}{l}\dot{q}_{conv} + \epsilon \sigma (T_{free}^4 - T_w^4) = \kappa (\frac{dT}{dx})_w\end{array} //]]>$

where  $//<![CDATA[ \begin{array}{l}T_{free}\end{array} //]]>$ is the free stream air temperature at the altitude of interest,  $//<![CDATA[ \begin{array}{l}T_w\end{array} //]]>$ is the wall temperature,  $//<![CDATA[ \begin{array}{l}\kappa = \frac{k}{\rho_0 C}\end{array} //]]>$ is the thermal diffusivity which uses  $//<![CDATA[ \begin{array}{l}k\end{array} //]]>$the thermal conductivity,  $//<![CDATA[ \begin{array}{l}\rho_0\end{array} //]]>$ the material density, and  $//<![CDATA[ \begin{array}{l}C\end{array} //]]>$ 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:

$//<![CDATA[ \begin{array}{l}\dot{q}_{conv} = C \rho^N V^M\end{array} //]]>$

where  $//<![CDATA[ \begin{array}{l}\dot{q}_{conv}\end{array} //]]>$ is the heat flux in W/cm^2, 

$//<![CDATA[ \begin{array}{l}C = (1.83\times 10^{-8}) \frac{1}{\sqrt{r_n} }(1-g_w)\end{array} //]]>$

is a constant that depends on the nose cone tip radius (in meters), and   $//<![CDATA[ \begin{array}{l}g_w\end{array} //]]>$ 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),  $//<![CDATA[ \begin{array}{l}\rho\end{array} //]]>$ is the free stream air density (in kg/m^3) at the flight altitude, and  $//<![CDATA[ \begin{array}{l}V\end{array} //]]>$ 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: