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Optimising the heat shield of the Space Shuttle

Optimising the thickness of the Space Shuttle's heat shield, using MATLAB to perform finite element analysis. Data for the external surface temperature of the shield was automatically extracted from .jpg graphs and translated into numerical data. The data was then used to simulate the energy transfer inside the shield with the Heat Equation, and various numerical methods were used. The shooting method was then applied to obtain the thickness of the shield from the maximum temperature at the inner surface.

ARC-1976-AC76-1713_orig.jpg

Extracting the data

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The data came in the form of a set of graphs of the temperature over time at different points on the shield. The paper from which this data was obtained is available here. A MATLAB function was created to automatically extract numerical data from the plots, which works as follow:

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  • The function reads the coordinates of the red pixels, and averages the y values for each x step.

  • Linear interpolation ensures no data points are missing.

  • The number of black pixels for each step of x and y is found, giving the coordinates of the origin at the maximums.

  • The coordinates of the last black pixels at y=0 and x=0 give the size of the axes.

  • The data is scaled, as the lengths of the axes the known and constant, and converted into °C.

502_extraction.jpg

Data from the paper for tile 502, with the amount of black pixels illustrated for each axis. 

502_extracted2.png

Data extracted from the .jpg image for tile 502.

Finite Element Analysis

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A 1D model was created to simulate the temperature throughout the tiles, with many possible variables such as the thickness of the tile, the simulation time of the number of steps. Different methods, such as the Forward Differencing method, the Dufort-Frankel method, the Backward Differencing method and the Crank-Nicolson method were implemented. A Neumann boundary was implemented at the inner surface.

Temperature profile of the tile 502 for a simulation time of 4000 s and a thickness of 0.05 m.

Shooting Method

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To optimise the thickness of the shield for a desired temperature at the inner surface, the shooting method was implemented. The function is ran for a two thickness guesses and calculate the maximum temperature reached at the inner surface. The difference from the aim is then used to calculate a new guess, and the process is repeated for as many iterations as desired. Below are the parameters and result for a maximum inner surface temperature of 175 °C:

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  • Tile n° = 502

  • Simulation time = 4000 s

  • Time steps = 501

  • Space steps = 21

  • Method = Crank-Nicolson

  • Aim = 175 °C

  • Guess 1 = 0.01 m

  • Guess 2 = 0.08 m

  • Iterations = 10

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  • Result = 0.0553 m

thickness_175°C.png

Temperature profile at the inner surface of tile 502, for a thickness of 55.3 mm

2D Modelling

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A 2D model of the tile was created, to simulate the temperature changes in its cross-section. Below is a simulation using the Dufort-Frankel method for tile 502, with a thickness of 0.05 m and a length of 0.15 m (length of a standard space shuttle tile). Here, the tile is perfectly insulated at on face and receives a uniform heat distribution on all other faces:

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2D modelling of tile 502 for a thickness of 0.05 m and length of 0.15 m.

Report

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