- #1
Xian
- 25
- 0
So a sadistic friend of mine posed this question:
Of course this question can be thought of in two ways: a) would the surface still be hot enough at the bottom or b) would the interior be hot enough at the bottom (since the coals would probably break, its a relevant question).
At the time I couldn't figure out how to model the problem, but now here's my approach:
Model the coal C as some compact 3-dimensional subset of R^{3}. Assume that the wind relative to the ground is negligible, that the air temperature is consant, and that the initial interior temperature of the coal is uniform. Let h be the height of the building, let \alpha be the thermal diffusivity of the coal, let T_{air} be the temperature of the air, and let T_{0} be the initial temperature of the coal.
Here's a big assumption, since the air in contact with the coal will be replaced by fresh air at temperature T_{air} as the coal falls, we will model the temperature of the falling coal by the heat equation with steady state boundary conditions. More specifically we will have T(x,y,z,0)=T_{0} in C and T(x,y,z,t)=T_{air} on the boundary \partial C and let the temperature evolve for a duration of \sqrt{2h/g} (the fall time) under the heat equation, \frac{\partial T}{\partial t}-\alpha\nabla^2 T = 0.
Is this a good model? I can already see where a few problem would arise. First of all, I'm concerned that the mathematics of heat transfer between two materials is more subtle than I assumed. Moreover, I'm worried that my assumption that air rushing past the coal can be treated as a steady state boundary condition is also wrong. If anything though, I think that final temperature distribution for the coal under this model would underestimate the real life solution at all points (is this fair?). I look forward for any comments, and eventually look to try to solve this for specially shaped coals (cubes, spheres, etc).
Of course this question can be thought of in two ways: a) would the surface still be hot enough at the bottom or b) would the interior be hot enough at the bottom (since the coals would probably break, its a relevant question).
At the time I couldn't figure out how to model the problem, but now here's my approach:
Model the coal C as some compact 3-dimensional subset of R^{3}. Assume that the wind relative to the ground is negligible, that the air temperature is consant, and that the initial interior temperature of the coal is uniform. Let h be the height of the building, let \alpha be the thermal diffusivity of the coal, let T_{air} be the temperature of the air, and let T_{0} be the initial temperature of the coal.
Here's a big assumption, since the air in contact with the coal will be replaced by fresh air at temperature T_{air} as the coal falls, we will model the temperature of the falling coal by the heat equation with steady state boundary conditions. More specifically we will have T(x,y,z,0)=T_{0} in C and T(x,y,z,t)=T_{air} on the boundary \partial C and let the temperature evolve for a duration of \sqrt{2h/g} (the fall time) under the heat equation, \frac{\partial T}{\partial t}-\alpha\nabla^2 T = 0.
Is this a good model? I can already see where a few problem would arise. First of all, I'm concerned that the mathematics of heat transfer between two materials is more subtle than I assumed. Moreover, I'm worried that my assumption that air rushing past the coal can be treated as a steady state boundary condition is also wrong. If anything though, I think that final temperature distribution for the coal under this model would underestimate the real life solution at all points (is this fair?). I look forward for any comments, and eventually look to try to solve this for specially shaped coals (cubes, spheres, etc).