for the balloon to float, what is the minimum temperature tmin of the hot air inside it? This is a topic that many people are looking for. cfiva.org is a channel providing useful information about learning, life, digital marketing and online courses …. it will help you have an overview and solid multi-faceted knowledge . Today, cfiva.org would like to introduce to you Estimate the average temperature of the air inside a hot air balloon 1-15. Following along are instructions in the video below:
Need the average temperature of the air inside the balloon and we have the mass mass of the unfilled balloon and the payload okay well at least. The first step know that the buoyancy. The buoyant force has to at least balance the weight.
So the volume f. Buoyancy. Yeah.
Right so the buoyancy force has to be at least equal to the weight or great greater than the weight. Lets just say equal so that means. Whats f buoyancy.
Thats the density of the surrounding air lets just call it rho 1 times. The volume times. The gravity right and that has to equal the mass m.
Plus. The mass of the air times gravity. So thats what we need first yeah go on vici and v.
Here is the volume of the balloon because it will displace because it will displace air equivalent to its weight because we know that not a mass equal density times volume okay and so if i multiply the volume of the balloon. Which will be the volume of the air displaced by the balloon outside times. The density of the balloon.
Which is we call the throw one so this is rho of the one sorry rho of the air surrounding air density of surrounding. Here. Because thats where this is taking place times.
Gravity. That should equal to the gravitational force. So.
This is the first thing. We need yeah. Okay now we need to relate this to temperature.
Okay. So small m. Thats small m.
Is the this is the mass of the air inside the balloon. We know that the mass of air inside the balloon is equal to the rho on the inside. Im gonna call it roll.
I just to distinguish it from the outside of the balloon actually im gonna call this rho all meaning outside the balloon and this is rho. I inside the balloon so rho. I times the volume of the balloon.
Well give me the mass of the air inside the bond. So im gonna take this and plug it here so we called this surrounding. Yeah okay so then were gonna get row.
0. Vg is equal to m. Plus.
We could cross out the gs. So. This is gonna be row.
I times v. Okay. So this is the first and then we could move this here.
So we get v and minus. I or yeah we could divide by v. So we get row outside minus row.
Inside is equal to m over v. Okay. So this is one equation.
We know we have to use the ideal gas law to relate the densities to the temperature yeah to the most so yeah. Let me. See how this is gonna fit.
So pv is nrt right and for air. Well i could play with this equation. And the number of moles is just the mass over the molar mass.
The molar mass is just call it mmm and rt if i divide both sides by or both sides by v. I would get p times. The molar mass equal ya mass over volume rt and this is nothing.
But the density. So now i have the pressure times. The molar mass equal the density times rt okay great so we already know the molar mass for air.
So that means the density of the air outside will just be the pressure times. The molar mass divided by rt sounds good and let me look up what the sir. The molar mass for a mole of air.
So here. This is the row for the air outside and with the same exact token for the inside will be the pressure. Which will be the same you know you have to have the same pressures right at the edge equilibrium times.
The molar mass. So this here would be 40 outside. If this is the density of the air outside and this here would be inside further than the temperature inside this is the density inside okay and then were gonna take these two and plug them into equation.
Lets call this equation two so plug in okay great and were gonna get p. Im gonna since these two are the same with the our im gonna factor them out so were gonna get p. Times.
The molar mass of air divided by r times. One over t outside minus. One over t inside right.
That has to equal em over v. Okay. Good yeah.
I got an equation and the only unknown in this equation as t inside because i can estimate everything else pressure. So yeah i looked at up the diameter was 15. Which is 15 16.
Lets just say that the radius is about 8 meters on a sphere approximation this would be fourth. There okay so if you want to go if youre gonna go little bit less than then a sphere well. Its a little bit more.
Thats why i kind of added a bit to the radius. Yeah. So lets see so okay.
So then well blow up the radius of the bed. Pi. R.
Cube. So this is 4. 3 pi.
8q lets see. 2145. 2800.
Yeah. I mean more than eight meters under a ds. So it should be okay.
Thats fine. Its its an approximation let me let me use what this based on a sphere calculation. Lets see what we got 2145 meter.
Cube. Okay. So.
This is the volume of the balloon right temperature outside is what calvin. Yeah. That should be zero.
No why lets just lets just take the the yeah. The average temperature of air. You see what we get usually 20 degrees more volume does not come to my pleasure.
Yeah okay so were going to take the temperature to be 290 and then the pressure and that altitude is about 10. 1. Times.
10 to the 5th. Thats well i need it in the equation yeah. 10.
To the 5. Then without sin pascals and i still have the our our is universal so gas constant thats 831. The units on that are joules per mole kelvin.
Okay thats it and then we just were just gonna solve for the t outside from this red equation. Here so. Lets see 1 over t.
Rt inside t. Inside. Lets just move everything else to the left.
Oh yeah. One more thing big. The molar mass of air.
We said is that yeah. 28. Point 96 grams.
Twenty eight point 96. So thats thats two point nine zero times. Ten to the minus two kilograms thats for air average.
Okay we have everything and big is given in the question. Thats the mass. Without thats the thats the mass of the unfilled balloon and the payload.
So thats 500. Okay. Now lets take this red equation.
Here and work it out for t inside. So we get one over t. Outside minus.
One over t inside equal r. And divided by p. Molar mass v.
Move. This to the right. And so one over t finished inside will be one over t 0.
Minus r. Over p. Molar mass v and then i could just reciprocate this so t.
I will be all that to the minus one and so lets plug the numbers. N we. Get 1 over.
2 19. Minus r. Is.
83. 1. Times.
500 divided by 10. 1. Times.
10 to the 5th molar mass. Is two point ninety times. 10.
To the minus two and the v. Is 2145 interviewed. Okay and this whole thing is raised to the minus one and let me see what we get so plug this into the calculator.
I got three hundred and fifty eight point eight for the temperature inside the balloon point eight calvin that sounds like a very reasonable temperature find the mass. Okay the mass of the air inside the balloon. Once we know the temperature that should become straightforward so we know from the ideal gas law.
Pv equals nrt and we already manipulated this to get this is m is mass over the molar mass of air rt. We divide by v. So we get p times.
The molar mass equal mass over volume is just the density inside yeah. But im interested in math. I want us to go away because im rt okay and if were talking about inside the balloon.
The mass. So this is the mass of air inside the balloon. This has to be inside the balloon.
This is the volume of the balloon and the pressure is the same so we get em i should be p. And. V rti and so that.
s. 10 12. V we said that this was 29 no mass of.
Air the volume of the balloon like this by our 831. And the temperature inside. We just found 358 kelvin.
Okay plugging this 3107 the simplest way to write this to do this is to start with the buoyancy force balancing the gravitational force yeah and then go from there basically what i did if you wanna know in summary is i balanced the gravitational force with the buoyancy force buoyancy force has in it the density of the air outside and the volume of the balloon. Partial mass has ended the density of the air times the volume of the balloon and from there on i related between the density outside and the density inside with the volume of the balloon and then i used the ideal gas law to express the densities in terms of the other variables. Pressure.
The molar mass of the air. Because we know that gas constant and the temperature in the case of density outside temperature outside in case for density inside temperature inside and thats how i was able to get from the density to the temperature. So i substituted.
I replaced densities with expressions having temperatures in them and then i plugged them in i know everything else and i solved for the temperature. In fact since everything else is no pressure volume of the balloon. We approximated the molar mass of the gas.
The universal gas constant the temperature outside all these variables. We can approximate or we have good numbers on here and we figured out the temperature inside and once we know the temperature inside. Its just a matter of plugging that back into the ideal gas law to figure out the mass of the air inside because now we have the temperature inside so i just used the ideal gas law inside.
I know the pressure has to be the same and by the way. Thats an argument. I use on the first part too because the pressure is the same inside and outside for equilibrium to happen and so so ya and that gives us the mass of d of the air inside the balloon.
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