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# Hot air balloons

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the science and maths behind balloon flight.. I took an interest in this after a balloon flight accross west auckland

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### Hot air balloons

1. 1. Hot Air Balloons The science and maths http:// johnwest.edublogs.org / Science Infoblog: A school science blog
2. 2. First a few useful definitions (fluids liquids and gasses) <ul><li>Fluids are substances that flow. </li></ul><ul><li>By definition all liquids and gasses are fluids. </li></ul><ul><li>Liquids are fluids that take the shape of the container they are stored in. </li></ul><ul><li>Gasses are fluids that expand to fill up the volume of their container. </li></ul>
3. 3. Flotation and Archimedes <ul><li>Archimedes of Syracuse was a physicist, engineer inventor and astronomer in his spare time. </li></ul><ul><li>…… and also probably the greatest mathematician ever. </li></ul><ul><li>Archimedes was killed by a Roman soldier during the siege of Syracuse c. 212 BC </li></ul><ul><li>Any object wholly or partially immersed in a fluid is buoyed up by a force equal to the weight of fluid displaced by the object (Archimedes Principle) </li></ul>Archimedes on wikipedia
4. 4. Generating Lift With Hot Air <ul><li>Raising the air temperature inside the envelope reduces its density. </li></ul><ul><li>The buoyant force according to Archimedes Principle is the weight of cold air displaced by the balloon. </li></ul><ul><li>This is greater than the force of gravity on the heated air </li></ul><ul><li>Hot air rises, its lighter than cold air </li></ul>Gravity Buoyant force
5. 5. Balloons have to be big to lift. How hot is the air inside? <ul><li>It is the combination of size and temperature difference between the hot air in the envelope and the ambient air temperature that determines the amount of lift. </li></ul><ul><li>If you want to design a balloon that will lift a basket with eight people in it…… how do you do the maths? </li></ul><ul><li>The ideal gas equation is a good start </li></ul>
6. 6. What do we know about gasses <ul><li>Gasses expand in a predictable way when they are heated under a constant pressure ( Charles’s law ) </li></ul><ul><li>The volume of a gas decreases when it is put under pressure ( Boyle’s law ). If you double the pressure on a gas its volume halves. ( as long as the temperature doesn’t change) </li></ul><ul><li>… .. And finally the volume of a gas only depends on the number of particles it contains (not so obvious)….see the next slide </li></ul>
7. 7. Counting atoms and molecules <ul><li>Air consists mostly of oxygen and nitrogen molecules </li></ul><ul><li>They are too small to count individually </li></ul><ul><li>Banks count out large numbers of coins by weighing them. They are bundled up in bags </li></ul><ul><li>The bundle of molecules that chemists work with is called a mole . </li></ul><ul><li>A mole of molecules contains 6x10 23 particles </li></ul><ul><li>The number of moles is represented in equations with the letter n </li></ul>
8. 8. The Ideal gas equation <ul><li>Combing what we know about gasses (boyle’s Law, Charles’s Law and the number of particles present) leads us to the Ideal Gas equation </li></ul>Increase p, V must decrease What happens to the gas when T increases? The gas constant,
9. 9. The Gas Constant <ul><li>If we measure pressure (p), volume (V), temperature (T in degrees absolute) and carefully weigh out an exact number of particles ( remember the bundle called a mole) then we can experimentally determine the value of R </li></ul>or
10. 10. Using the ideal gas equation to calculate air densities <ul><li>Hot air rises because it is less dense than surrounding air </li></ul><ul><li>We need to calculate air densities at different temperatures. We can rearrange the ideal gas equation to help us </li></ul><ul><li>It is easier to do this if we fiddle with the gas constant but it does mean introducing another term. The Molar Mass. </li></ul>
11. 11. Molar Mass <ul><li>Chemists need to count accurately the numbers of particles in a chemical reaction otherwise there is no point to their chemical equations </li></ul><ul><li>As said previously the bundle of particles they count with is called a mole (6.023X10 23 ) </li></ul><ul><li>A mole of hydrogen molecules weighs 2g </li></ul><ul><li>A mole of oxygen molecules weighs 32 g </li></ul><ul><li>A mole of water molecules weighs 18g </li></ul><ul><li>If you look at a periodic table it is easy to see where those numbers come from </li></ul>
12. 12. The gas equation and air density (1) <ul><li>The density of air over a range of temperatures can be calculated using the gas equation. </li></ul><ul><li>The number of moles of any gas present will be the mass of the gas divided by the mass of 1 mole. Now some maths. </li></ul>
13. 13. The gas equation and air density (2) <ul><li>The density of a gas is calculated by dividing its mass by the volume it occupies </li></ul><ul><li>Air density is represented by the Greek letter rho. </li></ul>
14. 14. The gas equation and air density (3) <ul><li>We need to substitute and rearrange the equation a bit </li></ul>Get rid of n Shift V to the other side
15. 15. The gas equation and air density (4) We can replace m/V with the air density and rearrange There is one last convenient trick
16. 16. The gas equation and air density (5). Using the specific gas constant <ul><li>The gas constant applies to any gas. We can tidy up the expression by using a new constant that is specific to a particular gas </li></ul>Finally…
17. 17. The final equation for calculating air density <ul><li>R (specific) is a constant </li></ul><ul><li>Pressure and temperature can be measured </li></ul>
18. 18. Balloon lift calculation <ul><li>P is the atmosperic pressure in Pascalls </li></ul><ul><li>R is the specific gas constant for air </li></ul><ul><li>T is the air temperature O K </li></ul><ul><li>The lift experienced by cubic metre of air at 95 o C is 0.245Kg </li></ul><ul><li>(ambient temp 15 o C) </li></ul>For typical atmospheric conditions 1/0.25839 or 4.073 m 3 of balloon needed to lift 1 Kg.