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# Modelling Conductive Heatflow Through A Heat-shielding Tile On A Spaceshuttle During Reentry

*Tutorhunt asked me to upload an article but I`m not sure the formatting on this transfers well to tutorhunt, if you`re actually interested I`ll happily email it to you, descri ption in the abstract.*

**Date **: 9/26/2016

Abstract

This report investigates different numerical methods for solving PDEs which are used to model the temperature change and inner surface temperature through a heat shielding tile, which is on a space shuttle re entering earths atmosphere. Matlab is used for forward differencing, backwards differencing, the Dufort Frankel Model, and Crank Nicolson Method. The stability of these methods is discussed and the efficiency of the most stable method is investigated. It was found that the Crank Nicolson is the most stable and accurate method. It gave the ideal thickness to be between 60 and 80mm, depending on it’s position the space shuttle.

Introduction

Numerical modelling of PDE’s is extremely important in designing real world systems where it is not always possible to measure there characteristics of a system. This could be due to it being extremely expensive, too large, or simply not feasible. In this report a space shuttle’s heat shielding tile is modelled as it is heated whilst re entering earths orbit. During re entry a space shuttle experiences extreme heating due to friction between the air and the shuttles body. The shuttle hits the earths atmosphere at hyper-sonic speeds 100km from the surface. The friction from the air causes it to heat to over 1000 °C causing almost anything to burn up, the pressure from this causes objects with low compressive strength to explode. (1) For the shuttle to avoid burning up it’s surface inside the tiles must not exceed 175°C (2).

Assumptions

⦁ The shuttle will always experience the same heating

The shuttles were reusable, and the heating effects from different ambient conditions, different entry speeds and angles was varied each time

⦁ The tile’s thickness is uniform

The tiles are assumed to have a constant cross section, and are not damaged.

⦁ The tile’s thermal properties are consistent.

The tiles were made batches, it’s assumed that each batch had the same thermal properties.

⦁ The tiles don’t degrade

The tiles had a long life time, it’s assumed that their properties didn’t change.

⦁ The tiles are only heated from one side

If there is any heating between the tiles it could cause them to behave differently. It’s assumed that they’re only heated from the external face.

Method

Four different methods of solving PDEs are used to solve the Fourier Equation which describes the heat flow through objects:Also required is the thermal diffusivity:

k

ρ

Cp

Forwards Differencing has first order accuracy in time, and second order accuracy in space due to truncation of the Taylor Series. It’s unstable when it’s timestep . This is very limiting and makes it one of the lesser used methods for solving PDEs.

The Dufort Frankel model is an unconditionally stable version of the leap frog method, it is accurate to the second order in both time and space, however at large timesteps experiences oscillatory behaviour.

The Backwards Differencing method approximates at the new timestep, calculating by looking back in time, this is the reverse of Forward Differencing.

This is more stable, it has first order accuracy in time and second order accuracy in space.

The Crank Nicholson method uses approximations of Forward and Backward Differencing methods.

This method has second order accuracy in time and space, and is stable at large timesteps.

Stability

The stability of the methods is found by increasing the time step and plotting the inner temperatures. The more consistent the temperatures are the more stable the method, when the temperature deviates suddenly it will be unstable at all larger step sizes.

Fig.1 Stability graph, the Crank Nicholson method is most stable, and Forward dDifferencing is least stable.

The results agree with the theory, with Forward Differencing quickly becoming unstable, the Dufort Frankel Model experiencing oscillatory behaviour at large time steps, Backwards Differencing and Crank Nicolson are stable, with Crank Nicolson having second order accuracy.

Results

The temperature data from the shuttle is found by converting a graph from NASA into a data vector on matlab. Tile 597 is used throughout this report. (3)

Fig.3 Temperature with time through the narrowest part of the tile.

This graph shows how the temperature inside the tile varies with time. The tile insulates the shuttle, as the inner surface is cooler than the outside. There is a thermal lag between the outside and inside, so the inner edge is still getting hotter when the outside is starting to cool down. This was also modelled in 3 dimensions, but the inner surface temperatures were identical. (appendix 1)

The temperature on the inside of the tile is found for different tile thicknesses, this should not exceed 175°C during re entry. The exact thickness required is found by using the shooting method to be 57mm. The bare minimum has no margin for error, so the inner surface temperature is found for a range of thicknesses.

Fig.4 Temperature for varying tile thicknesses, the thick purple line is the trendline from 100 thicknesses over the same range.

This shows an exponential change in maximum temperature for decreasing tile thickness. The greater the thickness the longer the delay before the maximum temperature is reached.

Efficiency

To illustrate how increasing the number of steps can cause a decrease in efficiency, the time to solve the problem was measured for a several numbers of time and spatial steps.

Fig.5 Graph of number of time and spatial steps against the time it takes to solve the problem.

Increasing either step size does not drastically impact the time taken, increasing both together can cause the program to become inefficient and take much longer requiring a lot more computing power. The final solution uses only 500 spatial and time steps, using more is unnecessary, but in larger models it may be unavoidable.

Discussion

For the final result 500 time and spatial steps were used, this is fast enough, and gives precise results. Using fewer steps works, but the difference in the time it takes is negligible.

The solution was modelled using both one and two dimensions, and it was found that although there’s a very small difference in some circumstances, normally there is no difference. This is because theat flows most quickly through the shortest path, so adding longer ones doesn’t change anything. However modelling a defective tile would need two or three dimensions. Real world conditions perhaps create the biggest innaccuracies, when space shuttles re entering earths will not experience the same heating effect every time. On a colder day air is denser, and there is more heating, whereas on a hot day the shuttle will fall faster through the less dense air, which would change the heating on the tile.

Conclusion

Numerical modelling is clearly a very good way of determining how something will behave in real world conditions. But it is very hard to base a design off modelling alone. Real world measurements are needed to validate the results. Of course in the case of a space shuttle this is very hard to do, and the 57mm tile thickness found by the shooting method would be a good guide. The designers would then have to consider how much safety factor to build into the shuttle. Adding extra thickness would make it a safer, as any damage it more likely to be on the surface and not affect the the rest of the shuttle. The real shuttle used thicknesses ranging from 25 to 127mm, so clearly there is not one correct thickness. (4) As real world validation is not available, modelling is the next best thing.

Modifications

⦁ Plottemp.m works automatically, and allows the user to select the temperature graph, first by cropping around the desired graph, then on the axes.

⦁ The program Thickness.m plots the temperature on the inside of the tile for a range of tile thicknesses, as well as a line through the maximum temperatures.

⦁ Timer.m creates a graph of the time taken by the program for varying number of steps. The data is saved at the end because it takes such a long time.

⦁ GUI2.m is a gui which runs plottemp and shuttle. It allows the user to change the variables and to select the temperature graph they want to use.

⦁ Shooting.m runs the shooting method to find the correct tile thickness for a desired inner surface maximum temperature.

⦁ Shuttle2D is exactly the same as shuttle, but instead uses two dimensions and plots the results changing in time. (6)

This resource was uploaded by: Sean