87 lines
2.2 KiB
TeX
87 lines
2.2 KiB
TeX
\documentclass{article}
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\usepackage[a4paper,left=2cm,top=2cm]{geometry}
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\usepackage{parskip}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{mathtools}
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\usepackage{hyperref}
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\begin{document}
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\title{Analysis of Superpressure Balloon using the ideal gas law}
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\author{Richard Meadows 2016}
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Analysis of Superpressure Balloon using the ideal gas law.
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\[
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P_{super} = P_{gas} - P_{air} \ \ \ \ (1)
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\]
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We can write the ideal gas equation for the gas inside the balloon:
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\[
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P_{gas}V = {m_{gas}\over{M_{gas}}} R T_{gas} \ \ \ \ (2)
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\]
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Since the system is floating, we know the mass of the air displaced is
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$m_{system}$. So we can also write the idea gas equation for the air
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displaced by the balloon.
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\[
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P_{air}V = {m_{system}\over{M_{air}}} R T_{air} \ \ \ \ (3)
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\]
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We assume the volumes are equal, so we can subsitiute one into the other.
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\[
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P_{gas} = P_{air} \bigg[ {{{m_{gas}\over{M_{gas}}} R T_{gas}}\over{{m_{system}\over{M_{air}}} R T_{air}}} \bigg] \ \ \ \ (4)
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\]
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Re-arrange and cancel $R$:
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\[
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P_{gas} = { P_{air} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } \bigg]} \ \ \ \ (5)
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\]
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Now we can use the definition of superpressure (1):
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\[
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P_{super} = P_{gas} - P_{air} \ \ \ \ (1)
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\]
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\[
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P_{super} = { P_{air} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } - 1\bigg]} \ \ \ \ (6)
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\]
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Substituting in our expression for $P_{air}$:
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\[
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P_{air} = {{m_{system}R T_{air}}\over{M_{air}V}} \ \ \ \ (3)
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\]
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\[
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P_{super} = { {{m_{system}R T_{air}}\over{M_{air}V}} \bigg[ { {m_{gas}T_{gas}M_{air}}\over{M_{gas}T_{air}m_{system}} } - 1\bigg]} \ \ \ \ (7)
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\]
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\[
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P_{super} = { {R\over{V}} \bigg[ { {m_{gas}}\over{M_{gas}} } T_{gas} - { {m_{system}}\over{M_{system}} } T_{air}\bigg]} \ \ \ \ (8)
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\]
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We define supertemperature in the same way as superpressure:
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\[
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T_{super} = T_{gas} - T_{air} \ \ \ \ (9)
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\]
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\[
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P_{super} = { {R\over{V}} \bigg[ \Big( {m_{gas}\over{M_{gas}}} - {m_{system}\over{M_{air}}} \Big)T_{air} + {{m_{gas}}\over{M_{gas}}}T_{super} \bigg]} \ \ \ \ \ \ (10)
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\]
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We can reasonably say the superpressure due to the temperature dominates, so
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\[
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{P_{super}\over{T_{super}}} \approx {{m_{gas}}\over{M_{gas}}}{R\over{V}} \ \ \ \ \ (11)
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\]
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\end{document} |