Page History

...

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
body T_{free}

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

Mathinline
body T_w

 is the wall temperature, 

Mathinline
body \kappa = \frac{k}{\rho_0 C}

 is the thermal diffusivity which uses 

Mathinline
body k

the thermal conductivity, 

Mathinline
body \rho_0

 the material density, and 

Mathinline
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.

...

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:

Mathinline
body \frac{dT}{dt} = \kappa \nabla^2 T = \kappa \frac{d^2 T}{dx^2}

and when this is solved the gradient $dT/dx$ can be found