balloon handout

8/11/2019 Balloon Handout

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1. First century of flight: ballooning

1.0 History of ballooning The twentieth century is often called “the century of flight”. But before the Wright brothers flew their Wright flyer in 1903, man ind had already been flying for 1!0 years,more than a century, using balloons. "or a long time #eo#le thought that this was the onlyway in which humans could fly.

The #rinci#le of hot air balloons was already nown to the $hinese in the year 300 %& asan in'ention by (huge )iang. The so*called +ong* ing lanterns were used #rimarily formilitary #ur#oses, #robably for communication #ur#oses. Today +ong* ing lanterns arestill widely used in %sia on festi'als and in many other celebrations. They are alsosometimes seen in -uro#e and are res#onsible for numerous /" re#orts .

+on* ing )antern

2ur#risingly enough, as far as we now, the $hinese ne'er considered to use a larger'ersion to lift a human. The first ones to seriously consider manned flight with a balloonwere two "rench brothers, ose#h* ichel ontgolfier and ac4ues*5tienne ontgolfier.

6n the end of the year 178! they started e #erimenting with larger hot air balloons.The first results were im#ressi'e. The lift force that could be generated by increasing the'olume of the hot air balloons was larger than they had e #ected. $onsider the effect of

scale, of magnification on the ratio of 'olume gas, lift andarea balloon surface, hence the weight .

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6n 1783 the ontgolfier brothers continued their e #eriments. The first li'ing creatures tolift off in a man*made de'ice were a shee#, a rooster and a duc . This was not a randomcollection of a nearby farm. 6t was #art of a scientific e #eriment to e #lore whetherhumans would be able to breathe and sur'i'e a flight higher in the atmos#here, away

from the earth surface. Birds that were able to fly could do this, so the duc was a controlsam#le to see whether the craft itself #erha#s had an effect. Whether it was a matter ofaccommodation or a difference between the #hysiology of birds and mammals had to betested by including a rooster, a bird that did not fly, and the shee#, as a mammal andmodel human. %ll animals sur'i'ed the flight in good health. Well, the rooster had a bro en wing, because it had been ic ed by the shee#

These e #erimental flights had by now drawn animmense attention of the crowd and the nobility.-'en the ing, )ouis :;6 witnessed the balloonflight with the animals. The ne t ste# was to ha'e

humans on board. "or this the ing #ro#osed to use #risoners as test sub<ects. =owe'er, there weresome noblemen who 'olunteered to be the firsthumans to fly. ne of them was >il?tre de @oAier. a! year old #hysician, who conducted a series oftethered flight e #eriments before the first real freeflight on Co'ember !1st, 1783. This first real, because untethered, flight was done together withan army officer, the ar4uis dD%rlandes.

>il?tre de @oAier continued as a flight #ioneer. =e set a record for s#eed 70 mEhr ,altitude 3000 m and distance tra'elled F! m in 178G. 6n 178F >il?tre de @oAier diedwhen he crashed during an attem#t to cross the -nglish $hannel.

Many other pioneering flights and new designs followed. Here you can see an experiment by Guyton de Morveau:a balloon with rudders attached to the craft to control thedirection of flight. Why doesn’t this work !"nd hencedon’t we see this today on balloons#

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Today hot air balloons are still called ontgolfiHres and the hybrid of a gas and hot air balloon is called a @oAiHre, after its in'entor who used hydrogen for the first time.

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1.1 Equation of state instead of gas law

1.1.1 Why we need the equation of stateBelow is the ideal gas law. But not in a 'ery useful form for aeronautics because it a#llies

to one amount of gas n mole with a 'olume of ; . Both n and ; are hard to define if yous#ea about changes of #ressure and tem#erature in the atmos#here.

3mol+ 8.31GF

p $ n %

with× = ×ℜ×ℜ =

Therefore, we will rewrite the ideal gas law to a different form. This new form is calledthe e&uation of state . The ad'antage is that it does not ha'e the 'olume or n in it. Byusing the air density instead, we can use it to describe any point in a flow or in theatmos#here. With the 'olume that would be much harder.

3mol+ 8.31GF

p $ n %

with× = ×ℜ×ℜ =

→!87.0 '

kg (

p ) %

with ) for air ρ = × ×=

1.1.2 Derivation of the equation of state from ideal gas law2ince we normally in the atmos#here do not now the 'olume$ nor the number of molesn, we want to get rid of the 'olume$ andn, and use the density instead. This density isdefined asI

I

m m$ $

mand n mass divided by mass per mole

M

ρ ρ = ⇒ =

=

Cow we re#lacen and$ by these e #ressionsI

!87 , 0.0!897kg ' kg ( mol air air

air

p $ n % substitute$ and n

m m p % divide left and right side by m

M

p% multiply by

M

p % redefine ) ) with M M M

ρ

ρ ρ

ρ

× = ×ℜ× ⇔

× = ×ℜ× ⇔

ℜ= × ⇔

ℜ ℜ= × × ⇔ = = = =

G



-4uation of stateI p ) % ρ = × ×

We will use this e4uation of state for many #ur#oses. But we ha'e to remember that the @ now has become de#endent on the molar mass of the gas. 2o the 'alue !87 is only 'alidfor air. We ha'e to ee# in mind the definition of @I

!87 , 0.0!897kg ' kg ( mol air air

air

) M

) with M M

ℜ=

ℜ= = =

F



/sing this nowledge we can calculate the lift force a balloon can generate.

The lift can be calculated using the difference between the weight of the balloon and theweight of the air that would ha'e been in this 'olume, had there not been a balloon.

Without the balloon, when there is air in this 'olume, there is an e4uilibrium. This meansthat the sum of all forces around this 'olume has to be e4ual to the weight of this air.These atmos#heric forces do not change when there is balloon in this 'olume, and theweight of the balloon is different from less than the weight of the air.

Therefore when there is a lighter craft li e a balloon in this 'olume, we can calculate thelift by sim#ly calculating the difference between the weight of the craft and the air itre#lacesI

1

G air gas

G atm gas

gasG atm

atm

, W W

, $ g $ g

, $ g

ρ ρ

ρ ρ

ρ

= −

= −

= − ÷

This true for both balloon ty#esI gas and hot air. But it is also not a 'ery #ractical form.@arely will we now the density of the gas or hot air.

"or a hot air balloon we would li e to now the lift de#ending on the increase oftem#erature and the 'olume. %nd for the gas balloon, we would li e to now the 'olumeas a function of the selected gas ty#e.

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1.3.2 !ift of hot"air balloons

)et us start with a hot air balloon. The interior is -% degrees hotter than the outside air in

the atmos#here. )et us use the abo'e e4uation with densities as a starting #ointI

1

I

1

1

1 1

1

hot air G atm

atm

hot G atm

atm

hot G atm

atm

atmG atm

hot

, $ g

and

p p ) %

) %

%his yields

p ) %

, $ g p

) %

% , $ g

%

% , $ g

%

ρ ρ

ρ

ρ ρ

ρ

ρ

ρ

= − ÷

= × × ⇔ =×

÷× ÷= − ÷ ÷×

÷ ÷= − ÷ ÷

= − ÷

ften it is more con'enient to use -% , the increase in tem#erature, com#ared to theoutside tem#erature,

1 atmG atm

atm

atm atmG atm

atm atm

% , $ g

% %

% % % , $ g

% % % %

ρ

ρ

= − ÷+ ∆

+ ∆= − ÷+ ∆ + ∆

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G atmatm

% , $ g

% % ρ

∆=

+ ∆

This is an e4uation that we can use. We now the density and tem#erature of theatmos#here. When we then need a certain lift we can calculate the re4uired tem#eraturefor a gi'en 'olume of the balloon. Then we can for e am#le see whether that is realistic,

tem#erature should be less than 1!0 M$ or whether we need a bigger balloon.

1.3.3 !ift of gas balloons

"or gas balloons we use the same e4uation as a start.

1 gasG atm

atm

, $ g ρ ρ ρ

= − ÷

6n hot air balloon the #ressure and molar mass wasthe same inside and outside the balloon. With gas balloons the #ressure and tem#erature are e4ual.

ne could argue that the #ressure inside might be alittle higher as a result of the tension of the balloonsmaterial. This is in the e4uations neglected below.6n larger balloons, li e weather balloons, the'olume of the balloon is much larger than the'olume of the gas when lifting off, to allow

e #ansion when reaching heights with a lower #ressure. 6n that case assuming an e4ual #ressure is acce#table.

/sing the assum#tion of e4ual #ressure and tem#eratures we can write for the ratio of thedensitiesI

1 1

! !

M p M p %

M % M ρ

ρ ρ ρ

ℜ ×= × × ⇔ = ⇔ =ℜ×

6f we substitute this in the e4uation abo'e, we getI

9

1 gasG atm

air

M , $ g

M ρ

= − ÷



To get a feeling for these numbers, here are the molar masses of some gassesI

=elium I =e N G.00 gEmol=ydrogenI =! N !.0! gEmol%ir a'eraged I air N !8.97 gEmol

2o for =elium the latter #art of the e4uation becomesIG.001 1!8.97 7

HeG atm atm atm

air

M , $ g $ g $ g

M ρ ρ ρ

= − = − ≈ × ÷ ÷ 6f we com#are this to hot*air balloonsI

G atmatm

% , $ g

% % ρ

∆= + ∆

This im#lies that -% should be si times as high as the% atm in +el'in. 2o for an outside

air tem#erature of 1F M$ !88 + , this would mean an air tem#erature a##roaching the1800 M$O This is hotter than a blowtorchO This clearly shows that in terms of generation olift, =elium is better. Which reasons can you thin of why the ma<ority of balloons useshot air des#ite this lower efficiencyJ

/sing the formulae we can calculate the lifting ca#acity #er cubic metre of gas at a'erageconditions. "or this we assume an air density of 1.!!F gEm3 and an air tem#erature of 1FM$ !88 + I

1 m3 =eliumIG.001 1.!!F 1.00 9.81 1 10!8.97

HeG atm

air

M , $ g

M ρ

= − = × × × − ≈ ÷ ÷

1 m3 1!0 M$ airI 10F1.!!F 1.00 9.81 3, !

!88 10FG atm

% , $ g .

% % ρ

∆= = × × × ≈+ ∆ +

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1.4 Balloon proble s

/roblem 0:Ty#ical 'olume for a 3*G #erson hot*air balloon is said to be !F00 m3. What is the total

weight of the balloon, bas et and #ayload for such a balloon, assuming the mentionedma imum tem#erature of 1!0 M$. assume *atm N 1.!!F gEm3, T N 1F M$, gN9.81 mEs!

/roblem 1:>arameters balloon of first human flight in a ontgolfiHre are said to beI 'olume 1700m3, lifting ca#acity 780 g Nballoon, bas et and #ayload, some claim an e'en higherweight of 830 g, but letDs now use the smaller figure of 780 g . /se the standard #ressure at sea le'elI 1013.!F mbar

a What was the tem#erature of the balloonJ %ssume an outside air tem#erature of10.0 M$.

b ther sources say the balloon was !3 m high and 1G m wide. $alculate the

tem#erature inside the balloon for this situation as well. =intI %ssume a s#hericalsha#e that is elongated in the 'ertical direction by a factor !3E1G .c Which source would you belie'e based on the outcomeJ What would you

estimate the 'olume of the real ontgolfiHre to ha'e beenJ

htt#IEE entlamberson.com

/roblem 2:=ow many helium filled #arty balloons would be re4uired to ta e*off for a #erson of 80

gJ % #arty balloon has a 'olume of about 1G liters. use *atm N 1.!!F gEm3

/roblem 3:When you get higher in the atmos#here, the air density becomes lower. The balloon wille #and because the #ressure is also lower. What will ha##en when the balloon reaches itsma imum altitudeJ What do you need to now to calculate that ma imum altitudeJ

/roblem 4:

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$onsider a gas balloon <ust after its release at sea le'el. The balloon does not changesha#e. This means on e'ery #art of the surface of the balloon the forces are ine4uilibrium, so a##arently the inside and outside force are e4ual. 2till, the o'erall effect isthat the balloon wants to accelerate to go u#. =ow is this #ossibleJ

/roblem 5:What are the ad'antages of a balloon com#ared to a winged aircraft in terms of liftJ %ndair dragJ 2o for what ty#es of flight is the balloon more efficient than the wingedaircraftJ

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