diff --git a/B-64-all.jpg b/B-64-all.jpg new file mode 100644 index 0000000..69543c6 Binary files /dev/null and b/B-64-all.jpg differ diff --git a/UBSEDL_2016-08-29T10-24-37_3.png b/UBSEDL_2016-08-29T10-24-37_3.png new file mode 100644 index 0000000..2554931 Binary files /dev/null and b/UBSEDL_2016-08-29T10-24-37_3.png differ diff --git a/UKHAS 2016 Conference References Extended.txt b/UKHAS 2016 Conference References Extended.txt new file mode 100644 index 0000000..8ae02ac --- /dev/null +++ b/UKHAS 2016 Conference References Extended.txt @@ -0,0 +1,38 @@ +Original refrences from Dan Bowen's 2011 talk ([01]-[14]), extended by Richard Meadows 2016 + +[00] University of Minnesota: Progress Report on High Altitude Plastic Balloons. 1952.pdf +[01] J. K. ANGELL and D. H. PACK, “ANALYSIS OF SOME PRELIMINARY LOW-LEVEL CONSTANT LEVEL BALLOON (TETROON) FLIGHTS,” MONTHLY WEATHER REVIEW, vol. 88, no. 7, pp. 235-248, Apr. 1960.pdf +[02] D. Booker and L. W. Cooper, “Superpressure Balloons for Weather Research,” Journal of Applied Meteorology, vol. 4, pp. 122–129, 1965. +[03] N. J. Cherry, “Characteristics and Performance of Three Low-Cost Superpressure Balloon (Tetroon) Systems,” Journal of Applied Meteorology, vol. 10, no. 5, pp. 982-990, 1971.pdf +[04] J. H. Hirsch and D. R. Booker, “Response of Superpressure Balloons to Vertical Air Motions,” Journal of Applied Meteorology, vol. 5, no. April, pp. 226-229, 1966.pdf +[05] W. H. Hoecker, “A Computer Program for Calculating Tetroon Inflation-Factor Nomographs,” Journal of Applied Meteorology, vol. 20, no. 8, pp. 949-954, 1981.pdf +[06] W. H. Hoecker, “A Universal Procedure for Deploying Constant-Volume Balloons and for Deriving Vertical Air Speeds from Them,” Journal of Applied Meteorology, vol. 14, no. September, pp. 1118-1124, 1975.pdf +[07] V. E. Lally and NCAR, “Superpressure Balloons for Horizontal Soundings of the Atmosphere,” NCAR, 0, 1967.pdf +[08] N. Levanon, R. A. Oehlkers, S. D. Ellington, and W. J. Massman, “On the Behavior of Superpressure Balloons at 150 mb,” Journal of Applied Meteorology, vol. 13, no. June, pp. 494-504, 1974.pdf +[09] P. Morel and W. Bandeen, “the EOLE experiments: early results and current objectives,” Bulletin of the American Meteorological Society, vol. 54, no. 4, pp. 298-304, 1973.pdf +[10] P. Morel, J. Fourrier, and P. Sitbon, 1968: The Occurrence of Icing on Constant Level Balloons. J. Appl. Meteor., 7, 626–634.pdf +[11] P. G. Scott, T. M. Lew, J. S. Wilbeck, J. L. Rand, and R. H. Brezinskv, “Long Duration Balloon Technology Survey,” Huntsville, 1996.pdf +[12] M. S. Smith and E. L. Rainwater, “OPTIMUM DESIGNS FOR SUPERPRESSURE BALLOONS,” Sulpher Springs, 2002.pdf +[13] TWERLE Team, “The TWERLE Experiment,” Bulletin of the American Meteorological Society, vol. 58, no. 9, pp. 936-948, 1977.pdf +[14] P. B. Voss, “Advances in Controlled Meteorological(CMET) Balloon Systems,” no. May. American Institute of Aeronautics and Astronautics, 1801 Alexander Bell Dr., Suite 500 Reston VA 20191-4344 USA,, pp. 1-5, 2009.pdf +[15] G. D. Nastrom: The Response of Superpressure Balloons to Gravity Waves. Journal of Applied Meteorology, 19, 1013–1019. 1980.pdf +[16] Geoffrey A. Landis: Low-altitude Exploration of the Venus Atmosphere by Balloon. 48th AIAA Aerospace Sciences Meeting, Orlando FL, January 6-9 2010.pdf +[17] M. KEIL, Met Office, Exeter, UK: Assimilating data from a simulated global constellation of stratospheric balloons. Q. J. R. Meteorol. Soc. 130, pp. 2475–2493. 2004.pdf +[18] THE CONCORDIASI PROJECT OVER ANTARCTICA DURING THE INTERNATIONAL POLAR YEAR (IPY). 2008.pdf +[19] Global Aerospace Corporation: Global Constellation of Stratospheric Scientific Platforms. Phase II Final Report. November 2002.pdf +[20] M. Pagitz and S. Pellegrino: Shape Optimization of “Pumpkin” Balloons. 2007.pdf +[21] L. A. Grass: Superpressure Balloon for Constant Level Flight. 1963.pdf +[22] JUSTIN H. SMALLEY: Balloon Shapes and Stresses Below the Design Altitude. December 1966.pdf +[23] Kumar et al.: DEVELOPMENT OF ULTRA-THIN POLYETHYLENE BALLOONS FOR HIGH ALTITUDE RESEARCH UPTO MESOSPHERE. 2014.pdf +[24] Frank Baginski, Michael Barg and William Collier: EXISTENCE THEOREMS FOR THIN INFLATED WRINKLED MEMBRANES SUBJECTED TO A HYDROSTATIC PRESSURE. 2006.pdf +[25] Henry M. Cathey, Jr. and David L. Pierce: Development of the NASA Ultra-Long Duration Balloon. 2007.pdf +[26] NCAR Technical Note 19: Low Modulus Strain Gages Stress Analysis of Balloon Structures. July 1966.pdf +[27] NCAR Technical Note 21, Harold L. Baker: Balloon Stress Band Analysis. September 1966.pdf + +Dan Bowen's 2011 talk: https://www.youtube.com/watch?v=jtfJuTvaHxo +Also Dan Bowen's 2012 talk: https://www.youtube.com/watch?v=cxkZViG4yoc&feature=youtu.be + +Richard Meadows's 2016 talk: https://www.youtube.com/watch?v=PQJAjDEq5AA&t=5h03m16s +https://github.com/richardeoin/a-quick-guide + +Vince Lally's papers are archived here: https://opensky.ucar.edu/islandora/object/archives%3Avinlally?display=list diff --git a/a-quick-guide.pdf b/a-quick-guide.pdf new file mode 100644 index 0000000..1def0c1 Binary files /dev/null and b/a-quick-guide.pdf differ diff --git a/a-quick-guide.tex b/a-quick-guide.tex new file mode 100644 index 0000000..dc8fe59 --- /dev/null +++ b/a-quick-guide.tex @@ -0,0 +1,367 @@ +\documentclass{beamer} +\usepackage[latin1]{inputenc} +\usepackage{hyperref} +\title[Small superpressure]{A quick guide to small superpressure} +\subtitle{\url{https://github.com/richardeoin/a-quick-guide}} +\author{Richard Meadows} +\institute{UKHAS Conference 2016} +\date{} +\begin{document} + +\begin{frame} + \titlepage +\end{frame} + + +\begin{frame}{Superpressure is.. } + + \begin{columns} + \begin{column}{0.6\textwidth} + \begin{itemize} + \item Gas sealed within the envelope. + %% if the balloon is to do anything useful, this gas will end + %% up at a higher pressure than the surrounding air - hence + %% the name + \item Envelope is intended to be inelastic. + %% that is, the envelope will stop stretching and become + %% stable, The resut of this is that the balloon remains at a + %% particular density-altitude. + + \end{itemize} + + \end{column} + \begin{column}{0.4\textwidth} + \begin{figure}[!ht] + %% image of lally balloon + \includegraphics[width=1\textwidth]{lally_1967_balloon.png} + \caption{GHOST Balloon, Lally 1967} + + %% this image is from when first + \end{figure} + \end{column} + \end{columns} + + +\end{frame} + +\begin{frame}{Can Amateurs do this too?} + + \begin{itemize} + \item Yes! + \item See also Dan Bowen at \href{https://ukhas.org.uk/general:ukhasconference}{UKHAS 2011}. + \end{itemize} + + \begin{columns} + \begin{column}{0.5\textwidth} + \begin{figure}[!ht] + %% ubseds6 + \includegraphics[width=1\textwidth]{ubseds6_altitude_plot.png} + \caption{UBSEDS6, 7th June 2015} + \end{figure} + + \end{column} + \begin{column}{0.5\textwidth} + \begin{figure}[!ht] + %% image of b-64 + \includegraphics[width=1\textwidth]{B-64-all.jpg} + \caption{B-64, Leo Bodnar 2014} + \end{figure} + \end{column} + \end{columns} + + % Multi-day flights with small envelopes (1-2 meters on the longest axis). + % Leo flight -- 134 days + + %% go back and check Dan's presentation too - I haven't got time to + %% return to everything he discussed. + +\end{frame} + +%% What does one look like in flight? + +\begin{frame}{In Flight} + + \begin{figure}[!ht] + %% image of ubseds20 + \centering + \includegraphics[width=0.8\textwidth]{UBSEDL_2016-08-29T10-24-37_3.png} + \caption{UBSEDS20 balloon at 12.5km float, 29th August 2016} + \end{figure} + + %% lots of people here contributed to this image.. + +\end{frame} +\begin{frame}{Floating} + + % Floating - what does this mean? + % calcualate density + + Float when: + + \[ + \text{Atmospheric Density} = \text{System Density} = {\frac{\Sigma{m}}{V}} + \] + + %% we can assume that the payload has no volume, and the same for + %% the material that makes the balloon. + + However, the balloon envelope stretches somewhat: + % Envelope isn't perfectly inelastic + + \[ + V = V_{initial}\times\Gamma + \] + + %% introduce gamma as ratio Vfloat / Vbuilt + + %% atmospheric density profile + \begin{figure}[!ht] + \centering + \includegraphics[width=0.8\textwidth]{isa_density_profile.png} + \caption{Density in the International Standard Atmosphere} + \end{figure} + + +\end{frame} +\begin{frame}{The Origins of Superpressure} + + %% Superpressure - where does this come from? + + \begin{itemize} + \item Free lift + %% more mols of gas inside than displaced outside + \item Supertemperature + %% aka. superheat, initial studies tend to use supertemperature, + %% so we'll stick with that. Floating greenhouse. + \item Vertical Air Currents (Lally 1967, VI. D. p.31) + %% less significant, < 10% + \end{itemize} + +\end{frame} + +\begin{frame}{Calculating Superpressure 1} + + Ideal gas law $PV = nRT$ + + \begin{columns} + \begin{column}{0.5\textwidth} + % gas + \begin{figure}[!ht] + \centering + \includegraphics[width=0.6\textwidth]{circle_gas.png} + \end{figure} + + \[ + P_{gas}V = {m_{gas}\over{M_{gas}}} R T_{gas} + \] + \end{column} + \begin{column}{0.5\textwidth} + % displaced air + \begin{figure}[!ht] + \centering + \includegraphics[width=0.6\textwidth]{circle_air_displaced.png} + \end{figure} + + \[ + P_{air}V = {m_{system}\over{M_{air}}} R T_{air} + \] + % can say this because we're floating + \end{column} + \end{columns} + + % now make volumes equal, and cancel R + +\end{frame} + +\begin{frame}{Calculating Superpressure 2} + + Definitions of Superpressure and Supertemperature: + % aka. superheat + + \[ + P_{super} = P_{gas} - P_{air} + \] + \[ + T_{super} = T_{gas} - T_{air} + \] + + Assuming volumes are equal: + + % taking the equation on the previous page, and after some algebra.. + % algebra is available as a separate document + \[ + P_{super} = { {R\over{V}} \bigg[ \Big( {m_{gas}\over{M_{gas}}} - {m_{system}\over{M_{air}}} \Big)T_{air} + {{m_{gas}}\over{M_{gas}}}T_{super} \bigg]} + \] + + % first term is due to extra gas - free lift, second due to supertemperature + + The second term dominates, so: + + \[ + {P_{super}\over{T_{super}}} \approx {{m_{gas}}\over{M_{gas}}}{R\over{V}} + \] + + % So superpressure and supertemperature are proportional - this is + % well known (Lally etc.) - and we want to minimise the constant of + % proportionality. + +\end{frame} + +% \item Effects of changing gamma. + + +\begin{frame}{Supertemperature} + + \begin{figure}[!ht] + %% lally table + \centering + \includegraphics[width=0.8\textwidth]{lally_19_table_9.png} + \caption{Lally 1967, Table 9 p.24 (edited)} + \end{figure} + + % this gives us a useful guesstimate at the supertemperature + +\end{frame} + +% I noted earlier that amateur balloons aren't spherical. Instead +% they're make flat and then inflated. Bristol SEDS, Leo, Qualatex +% are all essentially this shape. It's easy to make. + +\begin{frame}{Mylar Balloon Shape 1} + + % This is the "mylar balloon". + % shape. So called because mathematicians found this shape "in the + % wild" and named it after the object that took this shape - namely + % party balloons made from mylar. + + \begin{figure}[!ht] + %% mylar balloon shape + \centering + \includegraphics[width=0.7\textwidth]{paulsen_1994_figure_1.png} + \caption{Paulsen 1994, Figure 1} + \end{figure} + + \[ + \int_{0}^{a} \sqrt {1 + f'(x)^2}\ dx = r + \] + + % When you inflate it, the radius that the 2D shape had still + % exists. So it limits the shape + + % This is a well defined shape, can calcuate volume and so on - for + % instance the area of this cross section is 2 a^2 + +\end{frame} + +\begin{frame}{Mylar Balloon Shape} + + \begin{figure}[!ht] + \centering + \includegraphics[width=1\textwidth]{mylar_balloon_crimping_hot.png} + \caption{Crimping means a small area the in centre is stressed. } + \end{figure} + + %% The size of the area that's stressed is related to the + %% elasticisty of the material, which probably is quite low at + %% stratospheric temperatures. + + %% So this design doesn't appear to be much better than the tetroon, + %% where stress is concentrated at the corners. + + %% But we've got a trick... + +\end{frame} + +\begin{frame}{The Magic of Pre-stretch} + + %% Major step in making these balloons work - attributed to whom?? + + \begin{itemize} + \item Minimise Creep and relieve manufacturing stresses (Lally 1967, VI. C. p.28) + %% Lally knew about this + \item Increases $\Gamma$, leading to higher float and lower superpressure. + % our equation for density has volume on the bottom - we increase + % volume, get less dense and go higher. Same for pressure-thermal ratio + % Gamma ~1.7 for latest flights + \item Re-distributes stresses around mylar balloon shape. + %% When first built the stress is concentrated in the middle of each gore. + %% Pre-stretching equalises the stress over a much greater proportion of the gore. + + %% Pre-stretch generally good, as long as your material + %% mantains its properties. We haven't explored gamma > 2 regime + %% however. + + \end{itemize} + +\end{frame} + +\begin{frame}{Envelope Construction} + + \begin{figure}[!ht] + \centering + \includegraphics[width=0.9\textwidth]{bristol_seds_balloon_1_9m.png} + \caption{Drawing for 1.9m balloon} + \end{figure} + +\end{frame} + +\begin{frame}{Envelope Construction} + + \begin{figure}[!ht] + \centering + \includegraphics[width=1\textwidth]{bristol_seds_balloon_1_9m_film.png} + \caption{50$\mu$m film cross section} + \end{figure} + + Thanks to Exploratory Ideas grant from CEOI. + %% Paid for the lab time to take a look at this + +\end{frame} + +\begin{frame}{Further Work} + + \begin{itemize} + \item Web based calcuator - like the Burst Calculator. + \item Numerical analysis of previous flights. + \item Guidelines for minimum free lift. + %% drag equation + \item Modelling and measuring supertemperature. + %% not so easy, but do-able + \item Model for mylar tube shape. + %% bit of geometry + \item Explore $\Gamma > 2$ + %% the limit of pre-stretch + \item Measuring strain on the ground (Angell and Pack, Apr. 1960). + %% no specilist tools needed + \item Relationship between stress and strain. + %% in non-linear region - okay this is hard + + \end{itemize} + +\end{frame} + +\begin{frame}{Further Work} + + \begin{itemize} + \item Have fun flying round the world... + \end{itemize} + + \begin{figure}[!ht] + \centering + \includegraphics[width=0.6\textwidth]{pico-pi-logo.png} + \end{figure} + +\end{frame} + +\begin{frame}{Meridional Hoop} + + \begin{figure}[!ht] + \centering + \includegraphics[width=1\textwidth]{mylar_balloon_meridianal_hoop.png} + \caption{Meridional Hoop of a Mylar Balloon } + \end{figure} + +\end{frame} + + +\end{document} \ No newline at end of file diff --git a/a-quick-guide/a-quick-guide-0.png b/a-quick-guide/a-quick-guide-0.png new file mode 100644 index 0000000..a970cc0 Binary files /dev/null and b/a-quick-guide/a-quick-guide-0.png differ diff --git a/a-quick-guide/a-quick-guide-1.png b/a-quick-guide/a-quick-guide-1.png new file mode 100644 index 0000000..dde466d Binary files /dev/null and b/a-quick-guide/a-quick-guide-1.png differ diff --git a/a-quick-guide/a-quick-guide-10.png b/a-quick-guide/a-quick-guide-10.png new file mode 100644 index 0000000..b8fcc78 Binary files /dev/null and b/a-quick-guide/a-quick-guide-10.png differ diff --git a/a-quick-guide/a-quick-guide-11.png b/a-quick-guide/a-quick-guide-11.png new file mode 100644 index 0000000..c00f249 Binary files /dev/null and b/a-quick-guide/a-quick-guide-11.png differ diff --git a/a-quick-guide/a-quick-guide-12.png b/a-quick-guide/a-quick-guide-12.png new file mode 100644 index 0000000..9f86782 Binary files /dev/null and b/a-quick-guide/a-quick-guide-12.png differ diff --git a/a-quick-guide/a-quick-guide-13.png b/a-quick-guide/a-quick-guide-13.png new file mode 100644 index 0000000..0e20e57 Binary files /dev/null and b/a-quick-guide/a-quick-guide-13.png differ diff --git a/a-quick-guide/a-quick-guide-14.png b/a-quick-guide/a-quick-guide-14.png new file mode 100644 index 0000000..05a233b Binary files /dev/null and b/a-quick-guide/a-quick-guide-14.png differ diff --git a/a-quick-guide/a-quick-guide-15.png b/a-quick-guide/a-quick-guide-15.png new file mode 100644 index 0000000..5947b70 Binary files /dev/null and b/a-quick-guide/a-quick-guide-15.png differ diff --git a/a-quick-guide/a-quick-guide-16.png b/a-quick-guide/a-quick-guide-16.png new file mode 100644 index 0000000..54b667e Binary files /dev/null and b/a-quick-guide/a-quick-guide-16.png differ diff --git a/a-quick-guide/a-quick-guide-2.png b/a-quick-guide/a-quick-guide-2.png new file mode 100644 index 0000000..d9c81f7 Binary files /dev/null and b/a-quick-guide/a-quick-guide-2.png differ diff --git a/a-quick-guide/a-quick-guide-3.png b/a-quick-guide/a-quick-guide-3.png new file mode 100644 index 0000000..62b8353 Binary files /dev/null and b/a-quick-guide/a-quick-guide-3.png differ diff --git a/a-quick-guide/a-quick-guide-4.png b/a-quick-guide/a-quick-guide-4.png new file mode 100644 index 0000000..853216e Binary files /dev/null and b/a-quick-guide/a-quick-guide-4.png differ diff --git a/a-quick-guide/a-quick-guide-5.png b/a-quick-guide/a-quick-guide-5.png new file mode 100644 index 0000000..8574071 Binary files /dev/null and b/a-quick-guide/a-quick-guide-5.png differ diff --git a/a-quick-guide/a-quick-guide-6.png b/a-quick-guide/a-quick-guide-6.png new file mode 100644 index 0000000..82f09b0 Binary files /dev/null and b/a-quick-guide/a-quick-guide-6.png differ diff --git a/a-quick-guide/a-quick-guide-7.png b/a-quick-guide/a-quick-guide-7.png new file mode 100644 index 0000000..b0db213 Binary files /dev/null and b/a-quick-guide/a-quick-guide-7.png differ diff --git a/a-quick-guide/a-quick-guide-8.png b/a-quick-guide/a-quick-guide-8.png new file mode 100644 index 0000000..b89a0d4 Binary files /dev/null and b/a-quick-guide/a-quick-guide-8.png differ diff --git a/a-quick-guide/a-quick-guide-9.png b/a-quick-guide/a-quick-guide-9.png new file mode 100644 index 0000000..559239d Binary files /dev/null and b/a-quick-guide/a-quick-guide-9.png differ diff --git a/bristol_seds_balloon_1_9m.pdf b/bristol_seds_balloon_1_9m.pdf new file mode 100644 index 0000000..c59fb45 Binary files /dev/null and b/bristol_seds_balloon_1_9m.pdf differ diff --git a/bristol_seds_balloon_1_9m.png b/bristol_seds_balloon_1_9m.png new file mode 100644 index 0000000..3f92e74 Binary files /dev/null and b/bristol_seds_balloon_1_9m.png differ diff --git a/bristol_seds_balloon_1_9m_film.png b/bristol_seds_balloon_1_9m_film.png new file mode 100644 index 0000000..050e163 Binary files /dev/null and b/bristol_seds_balloon_1_9m_film.png differ diff --git a/circle_air_displaced.png b/circle_air_displaced.png new file mode 100644 index 0000000..41a0986 Binary files /dev/null and b/circle_air_displaced.png differ diff --git a/circle_gas.png b/circle_gas.png new file mode 100644 index 0000000..fd83ee9 Binary files /dev/null and b/circle_gas.png differ diff --git a/ideal_gas_analysis.pdf b/ideal_gas_analysis.pdf new file mode 100644 index 0000000..1bc9d33 Binary files /dev/null and b/ideal_gas_analysis.pdf differ diff --git a/ideal_gas_analysis.tex b/ideal_gas_analysis.tex new file mode 100644 index 0000000..f10f855 --- /dev/null +++ b/ideal_gas_analysis.tex @@ -0,0 +1,87 @@ +\documentclass{article} +\usepackage[a4paper,left=2cm,top=2cm]{geometry} + +\usepackage{parskip} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{hyperref} + +\begin{document} + +\title{Analysis of Superpressure Balloon using the ideal gas law} +\author{Richard Meadows 2016} + +Analysis of Superpressure Balloon using the ideal gas law. + +\[ +P_{super} = P_{gas} - P_{air} \ \ \ \ (1) +\] + +We can write the ideal gas equation for the gas inside the balloon: + +\[ + P_{gas}V = {m_{gas}\over{M_{gas}}} R T_{gas} \ \ \ \ (2) +\] + +Since the system is floating, we know the mass of the air displaced is +$m_{system}$. So we can also write the idea gas equation for the air +displaced by the balloon. + +\[ + P_{air}V = {m_{system}\over{M_{air}}} R T_{air} \ \ \ \ (3) +\] + +We assume the volumes are equal, so we can subsitiute one into the other. + +\[ + P_{gas} = P_{air} \bigg[ {{{m_{gas}\over{M_{gas}}} R T_{gas}}\over{{m_{system}\over{M_{air}}} R T_{air}}} \bigg] \ \ \ \ (4) +\] + +Re-arrange and cancel $R$: + +\[ + P_{gas} = { P_{air} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } \bigg]} \ \ \ \ (5) +\] + +Now we can use the definition of superpressure (1): + +\[ +P_{super} = P_{gas} - P_{air} \ \ \ \ (1) +\] + +\[ + P_{super} = { P_{air} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } - 1\bigg]} \ \ \ \ (6) +\] + +Substituting in our expression for $P_{air}$: + +\[ + P_{air} = {{m_{system}R T_{air}}\over{M_{air}V}} \ \ \ \ (3) +\] + +\[ + P_{super} = { {{m_{system}R T_{air}}\over{M_{air}V}} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } - 1\bigg]} \ \ \ \ (7) +\] + +\[ + P_{super} = { {R\over{V}} \bigg[ { {m_{gas}}\over{M_{gas}} } T_{gas} - { {m_{system}}\over{M_{system}} } T_{air}\bigg]} \ \ \ \ (8) +\] + +We define supertemperature in the same way as superpressure: + +\[ + T_{super} = T_{gas} - T_{air} \ \ \ \ (9) +\] + +\[ + P_{super} = { {R\over{V}} \bigg[ \Big( {m_{gas}\over{M_{gas}}} - {m_{system}\over{M_{air}}} \Big)T_{air} + {{m_{gas}}\over{M_{gas}}}T_{super} \bigg]} \ \ \ \ \ \ (10) +\] + +We can reasonably say the superpressure due to the temperature dominates, so + +\[ + {P_{super}\over{T_{super}}} \approx {{m_{gas}}\over{M_{gas}}}{R\over{V}} \ \ \ \ \ (11) +\] + +\end{document} \ No newline at end of file diff --git a/isa_density_profile.png b/isa_density_profile.png new file mode 100644 index 0000000..6254d8d Binary files /dev/null and b/isa_density_profile.png differ diff --git a/lally_1967_balloon.png b/lally_1967_balloon.png new file mode 100644 index 0000000..f81e347 Binary files /dev/null and b/lally_1967_balloon.png differ diff --git a/lally_19_table_9.png b/lally_19_table_9.png new file mode 100644 index 0000000..3dacf2f Binary files /dev/null and b/lally_19_table_9.png differ diff --git a/mylar_balloon_crimping.png b/mylar_balloon_crimping.png new file mode 100644 index 0000000..a145249 Binary files /dev/null and b/mylar_balloon_crimping.png differ diff --git a/mylar_balloon_crimping_factor.png b/mylar_balloon_crimping_factor.png new file mode 100644 index 0000000..9a36cb7 Binary files /dev/null and b/mylar_balloon_crimping_factor.png differ diff --git a/mylar_balloon_crimping_hot.png b/mylar_balloon_crimping_hot.png new file mode 100644 index 0000000..3157588 Binary files /dev/null and b/mylar_balloon_crimping_hot.png differ diff --git a/mylar_balloon_equatorial_hoop.png b/mylar_balloon_equatorial_hoop.png new file mode 100644 index 0000000..4c4993c Binary files /dev/null and b/mylar_balloon_equatorial_hoop.png differ diff --git a/mylar_balloon_grid.png b/mylar_balloon_grid.png new file mode 100644 index 0000000..130ae15 Binary files /dev/null and b/mylar_balloon_grid.png differ diff --git a/mylar_balloon_meridianal_hoop.png b/mylar_balloon_meridianal_hoop.png new file mode 100644 index 0000000..e490534 Binary files /dev/null and b/mylar_balloon_meridianal_hoop.png differ diff --git a/paulsen_1994_figure_1.png b/paulsen_1994_figure_1.png new file mode 100644 index 0000000..7af4833 Binary files /dev/null and b/paulsen_1994_figure_1.png differ diff --git a/paulsen_1994_limit.png b/paulsen_1994_limit.png new file mode 100644 index 0000000..4541baf Binary files /dev/null and b/paulsen_1994_limit.png differ diff --git a/pico-pi-logo.png b/pico-pi-logo.png new file mode 100644 index 0000000..a65fe28 Binary files /dev/null and b/pico-pi-logo.png differ diff --git a/ubseds6_altitude_plot.png b/ubseds6_altitude_plot.png new file mode 100644 index 0000000..81f3060 Binary files /dev/null and b/ubseds6_altitude_plot.png differ