"Hey Gary, you know we should fill airliners with Helium to reduce their weight, saving millions of gallons of fuel and possibly the planet!" Let's examine this idea a little more closely. Now, there ensued a lengthy debate about the required structure and air resistance, etc. However, my colleague suggested we wouldn't need to fill an entire airliner with Helium, just a portion of it. So, I suggested a caleulation. Consider an empty 747 filled with helium, how much buoyant force would there be? We have the following information: • Mass of empty 747 = 180,000 kg. • Interior volume of 747 = 1100 m. • Density of air (STP): Pair = 1.203 kg/m². Density of He (STP);pue = 0.1786 kg/m². a) Determine the buoyant force of the He-filled aircraft as it sits on the tarmac (on the ground). The buoyant force is determined by the weight of outside fluid displaced (here air) less the weight of fluid that is in the displaced volume (here Helium). Now let's examine the buoyant force at altitude. Typical cruise altitude is h = 12,000 m. The atmospheric density of the Earth can be modeled in the most basic way by the following differential equation, dp -kdh where h is the height above the surface and k is a constant to be determined. b) What is the solution of this differential equation? That is, find the function p(h). (Recall, pressure halves for every 5500 m.) c) The pressure at an altitude h = 5500m is one half that at the surface. Determine the constant k. d) Find the density of air and helium at the cruising altitude and calculate the buoyant force. (Assume the interior of the aircraft is at the same pressure as outside.)
"Hey Gary, you know we should fill airliners with Helium to reduce their weight, saving millions of gallons of fuel and possibly the planet!" Let's examine this idea a little more closely. Now, there ensued a lengthy debate about the required structure and air resistance, etc. However, my colleague suggested we wouldn't need to fill an entire airliner with Helium, just a portion of it. So, I suggested a caleulation. Consider an empty 747 filled with helium, how much buoyant force would there be? We have the following information: • Mass of empty 747 = 180,000 kg. • Interior volume of 747 = 1100 m. • Density of air (STP): Pair = 1.203 kg/m². Density of He (STP);pue = 0.1786 kg/m². a) Determine the buoyant force of the He-filled aircraft as it sits on the tarmac (on the ground). The buoyant force is determined by the weight of outside fluid displaced (here air) less the weight of fluid that is in the displaced volume (here Helium). Now let's examine the buoyant force at altitude. Typical cruise altitude is h = 12,000 m. The atmospheric density of the Earth can be modeled in the most basic way by the following differential equation, dp -kdh where h is the height above the surface and k is a constant to be determined. b) What is the solution of this differential equation? That is, find the function p(h). (Recall, pressure halves for every 5500 m.) c) The pressure at an altitude h = 5500m is one half that at the surface. Determine the constant k. d) Find the density of air and helium at the cruising altitude and calculate the buoyant force. (Assume the interior of the aircraft is at the same pressure as outside.)
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Answer to (b), (c) and (d) only, please.
Transcribed Image Text:"Hey Gary, you know we should fill airliners with Helium to reduce their weight, saving millions of gallons of fuel and
possibly the planet!"
Let's examine this idea a little more closely. Now, there ensued a lengthy debate about the required structure and air
resistance, etc. However, my colleague suggested we wouldn't need to fill an entire airliner with Helium, just a portion of
it. So, I suggested a caleulation. Consider an empty 747 filled with helium, how much buoyant force would there be? We
have the following information:
• Mass of empty 747 = 180,000 kg.
• Interior volume of 747 = 1100 m.
• Density of air (STP): Pair = 1.203 kg/m². Density of He (STP);pue = 0.1786 kg/m².
a) Determine the buoyant force of the He-filled aircraft as it sits on the tarmac (on the ground). The buoyant force
is determined by the weight of outside fluid displaced (here air) less the weight of fluid that is in the displaced
volume (here Helium).
Now let's examine the buoyant force at altitude. Typical cruise altitude is h = 12,000 m. The atmospheric density of
the Earth can be modeled in the most basic way by the following differential equation,
dp
-kdh
where h is the height above the surface and k is a constant to be determined.
b) What is the solution of this differential equation? That is, find the function p(h). (Recall, pressure halves for
every 5500 m.)
c) The pressure at an altitude h = 5500m is one half that at the surface. Determine the constant k.
d) Find the density of air and helium at the cruising altitude and calculate the buoyant force. (Assume the
interior of the aircraft is at the same pressure as outside.)
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