Undergraduate Research (AE 4699)
Final Report
Wagner, Killian C
12-11-2015
Introduction:
This semester was spent on the research, design, and development of sapphire windows which
would be used to transmit infrared beams onto a high-temperature, high-pressure burner in order to
perform spectroscopic analysis. The selection of the material for the lens of this material was sapphire
due to its ability to transmit efficiently at the required wavelength and its resistance to stresses and
scratches.
A portion of this semester was spent determining how to thick the window is required to be in
order to withstand the pressure that it will undergo during the course of experiments. This involved
doing research to see what equations needed to be set-up and then implementing the parameters
relevant to the experiment at hand.
The current window that needed to be replaced had dimensions of the following:
D = 3 inches = 7.62 cm
And the objective was to develop a designed window configuration and determine the
dimensions of the lens that would be associated with it as these had to be based on cost-effective.
Following this, companies were contacted and quotes were requested for lenses within a dimension
range as the window base would be developed to match the dimensions of whichever window lens was
deemed most cost effective. Lastly, another portion of this semester was spent developing the model of
the window and its frame in SolidWorks so that the drawings for each of the components with
dimensions could be developed and sent to in-house manufacturing in order to reduce cost of
development.
Development of above Plot:
D = Diameter of Window (measured)
T = Required Thickness of Window to sustain max pressure
To obtain this plot and solve for the required material thickness, we use the following equation for the
maximum stress in a loaded uniform, circular plate:
𝑆𝑚𝑎𝑥 =
𝐾𝐷 2 𝑃
4𝑇 2
Where
𝑆𝑚𝑎𝑥 =
𝐹𝑎
𝑆𝑓
Sf: the safety factor of the window, a selected parameter
Fa is the common apparent elastic limit for Al2O3 (sapphire) so that no permanent deformation results in
the window. It is a material property
𝐹𝑎 𝐾𝐷 2 𝑃
=
𝑆𝑓
4𝑇 2
Rearranging to solve for thickness, T
𝑆𝑓 𝐾𝑃
𝑇 = 𝐷√
4𝐹𝑎
Sf: the safety factor for the window (selected value of 4 for many laboratory applications)
K: a factor that depends on the support method for the window. In this case, it is unclamped so
K=Ku=1.1125
Once these values are substituted into T expression:
𝑃
𝑇 = 1.06066 𝐷√
𝐹𝑎
Consulting a material properties table gives us Fa
𝐹𝑎 = 276 𝑀𝑃𝑎 = 45000 𝑝𝑠𝑖
𝑃𝑚𝑎𝑥
𝑇𝑚𝑎𝑥 = (1.06066)𝐷√
45000
Pmax = 25 atm = 367.3975 psi
Current laboratory window has D = 3 in = 7.62 cm
𝑇𝑚𝑎𝑥 = 0.7303 𝑐𝑚
This thickness represents the minimum required thickness of the window at this selected
diameter to withstand the maximum pressure that will occur during the course of the experiment
without any permanent deformation.
For a sample MATLAB script to create the above plots, the code attached to end of the report.
For the clamped case:
The only variation compared to the unclamped case is that the value of K is now Kc, rather than Ku:
𝐾 = 𝐾𝑐 = .75
The expression for the thickness, T, becomes:
𝑆𝑓 𝐾𝑐 𝑃
𝑇 = 𝐷√
4𝐹𝑎
and the value of Sf stays the same as previously
𝑃
𝑇 = 0.866 𝐷 √
𝐹𝑎
And since we are still examining sapphire, Fa retains its value of 45,000 psi. The expression for Tmax that
can be written is:
𝑇𝑚𝑎𝑥,𝑐𝑙𝑎𝑚𝑝𝑒𝑑 = (0.866)𝐷√
𝑃𝑚𝑎𝑥
45000
Or as expressed in millimeters, the thickness vs. diameter plot becomes:
A window of 25.0 mm diameter with a thickness of 2.3 mm would be put in the region above
this curve meaning that the material properties of the window would be sufficient for the experiment at
hand. The safety factor for a window of these dimensions could be calculated by rearranging the
thickness formula as follows:
𝑆𝑓 𝐾𝑐 𝑃
𝑇 = 𝐷√
4𝐹𝑎
𝑆𝑓 =
4𝐹𝑎 𝑇 2
𝐾𝑐 𝑃𝐷 2
When substituting the specific values in:
𝑆𝑓 =
4(45000 𝑝𝑠𝑖)(2.3 𝑚𝑚)2
(0.75)(367.40)(25 𝑚𝑚)2
𝑆𝐹 = 5.5
This value was rounded down in order to generate a conservative estimate, but it is still well
above the selected value of 4, which is a common value in laboratory experiments.
SF: Safety Factor of the Window
Selection of Material:
The material that was selected as the lens for this window configuration was sapphire due to its
ability to resist wear, withstand high pressures, and transmit a large range of light wavelengths. In the
experiments to be performed, it was important that the material transmits at about 5 micrometers with
a high efficiency and the ability of sapphire to perform to this standard is given in the following plot:
An example of a graduate student performing a similar experiment and the means by which
they selected their lens material is given in the following link:
http://openscholarship.wustl.edu/cgi/viewcontent.cgi?article=1502&context=etd
Purchase Options:
Another portion of this semester was spent collecting data on the sapphire lens optics that are
available on the market. This would allow us to size the metallic pieces and develop the SolidWorks
representation of each of the pieces. This stage of the design process involved going online and
collecting as much information for these lenses as possible from multiple companies. For each possible
lens, the safety factor was calculated in order to determine whether the piece would meet lab safety
requirements (in many cases it did not). Below is the majority of the collected data as it was inputted
into an Excel spreadsheet:
Purchase Options from Edmund Optics:
Purchase Options from Esco Optics:
Purchase Options from Newport:
Complete data:
In this table, the Design column represents whether the configuration is clamped or unclamped
as this affects the safety factor calculation.
Final Window Design:
The window design is composed of two primary steel pieces and then the lens itself. The smaller
piece is what attaches to the experiment set-up itself. It is anchored to the larger metal piece by screws
which can be seen in the following image:
The larger holes in the big metallic circular section are for bolts that will lock the configuration
onto the burner set-up. The circular incisions in each of the metal frames is where an O-ring will be
placed in order to ensure that there is no contact between the metal surfaces. There is also a small divot
in each of metal pieces which is half the thickness of the lens (half of 3 mm) and this will ensure that the
lens is locked into place once the bolts are in place. The location of the lens is off center so that it can
capture the entire experimental area.
Final Requested Sapphire Window Lens:
Diameter: 1.5 inches
Thickness: 0.11811 inches = 3 mm
SF exceeds 4 experiment requirement
Code: