Is Exponential Form Better for Balloon with Inserted Load Calculation?

[erobz]

I don't think so, the results are different. Something is not right, the values are either too small or too large...

OK, thanks. So, is ##\rho_{out} - \frac{m}{V} = \rho_{out} \frac{T_0}{T_{in}} (1-\beta h)##, with ##\rho_{out} = \rho_0 (1 - \alpha h)##, correct?

I didn't process this ##\uparrow##

The equation is.

$$ \frac{ \rho_{in}V + m }{V} = \rho_{out} $$

How did you get what you did above?

 

[Hak]

I didn't process this ##\uparrow##

The equation is.

$$ \frac{ \rho_{in}V + m }{V} = \rho_{out} $$

How did you get what you did above?

I do not understand. Didn't we say we could match the average density ##\rho_{avg}## to the external density ##\rho_{out}##?

 

[erobz]

I do not understand. Didn't we say we could match the average density ##\rho_{avg}## to the external density ##\rho_{out}##?

Yeah, that it what I think is meant by the equation you quote? it is saying at the equilibrium position the average density of the entire balloon system will equal the density of the surrounding atmosphere at ##h##.

 

[Hak]

Yeah, that it what I think is meant by the equation you quote? it is saying at the equilibrium the average density of the entire balloon system will equal the density of the surrounding atmosphere at ##h##.

OK, then? I can't grasp the nettle...

 

[erobz]

OK, then? I can't grasp the nettle...

Show your work on that equation please, if it doesn't work out then so be it, but we need to see how you rearranged the equation.

 

[Hak]

All I can say about ##\rho_{in}## is: $$\frac{\rho_{in}}{\rho_{out}} = \frac{t_{out}}{t_{in}}$$. I cannot see the immediate dependence on ##h##.

I did not understand. Would you like me to show how I equated this equation above with the last one you quoted?

 

[erobz]

I did not understand. Would you like me to show how I equated this equation above with the last one you quoted?

Start with the equation in post #71 and show the algebraic manipulations and substitutions you performed, because when I do those, I get the same result we got earlier.

 

[Hak]

Start with the equation in post #71 and show the algebraic manipulations and substitutions you performed, because when I do those, I get the same result we got earlier.

I get:

$$\rho_{in} + \frac{m}{V} = \rho_{out} \Rightarrow \ \rho_{in} + \frac{m}{V} = \rho_{0} (1 - \alpha h)$$.

From ##\frac{\rho_{in}}{\rho_{out}} = \frac{T_{out}}{T_{in}}##, we get: ##\rho_{in} = \rho_{out} \frac{T_{out}}{T_{in}}##. Plugging in the previous one:

$$\rho_{out} \frac{T_{out}}{T_{in}} - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$. Finally:

$$\rho_{0} (1 - \alpha h)\frac{T_{0}}{T_{in}} (1 - \beta h) - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$.

 

[erobz]

I get:

$$\rho_{in} + \frac{m}{V} = \rho_{out} \Rightarrow \ \rho_{in} + \frac{m}{V} = \rho_{0} (1 - \alpha h)$$.

From ##\frac{\rho_{in}}{\rho_{out}} = \frac{T_{out}}{T_{in}}##, we get: ##\rho_{in} = \rho_{out} \frac{T_{out}}{T_{in}}##. Plugging in the previous one:

$$\rho_{out} \frac{T_{out}}{T_{in}} - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$. Finally:

$$\rho_{0} (1 - \alpha h)\frac{T_{0}}{T_{in}} (1 - \beta h) - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$.

Which is the same result you initially found ( applying Newtons Second )?

 

[Hak]

Which is the same result you initially found ( applying Newtons Second )?

Yes. I have discordant values, however. Now the two values come out to me as ##h \approx 32.6 \ m## and ##h \approx 11 \ km##. They are not convincing at all. Could you tell me what numerical result you get?

 

[erobz]

$$\rho_{0} (1 - \alpha h)\frac{T_{0}}{T_{in}} (1 - \beta h) - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$.

As you did before:

$$\rho_{0} (1 - \alpha h)\frac{T_{0}}{T_{in}} (1 - \beta h) - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$.

## \gamma = \frac{T_{0}}{T_{in}} ##

$$\rho_{0} (1 - \alpha h) \gamma (1 - \beta h) - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$.

Multiply everything by ( or divide by) ##-1## :

$$-\rho_{0} (1 - \alpha h) \gamma (1 - \beta h) +\rho_{0} (1 - \alpha h) - \frac{m}{V} = 0$$
or (rearranging)
$$ \rho_{0} (1 - \alpha h) -\rho_{0} (1 - \alpha h) \gamma (1 - \beta h) - \frac{m}{V} = 0 $$

Factor:

$$\rho_{0} (1 - \alpha h) ( 1 - \gamma (1 - \beta h) ) - \frac{m}{V} = 0 $$

Multiply both sides by ##V##:

$$\rho_{0} (1 - \alpha h) ( 1 - \gamma (1 - \beta h) )V - m = 0 $$

That is the same relationship I got! Multiply everything by ##g## and we are back at the working that used Newtons Second Law directly.

 

[Hak]

As you did before:

$$\rho_{0} (1 - \alpha h)\frac{T_{0}}{T_{in}} (1 - \beta h) - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$.

## \gamma = \frac{T_{0}}{T_{in}} ##

$$\rho_{0} (1 - \alpha h) \gamma (1 - \beta h) - \rho_{0} (1 - \alpha h) + \frac{m}{V} = 0$$.

Multiply everything by ( or divide by) ##-1## :

$$-\rho_{0} (1 - \alpha h) \gamma (1 - \beta h) +\rho_{0} (1 - \alpha h) - \frac{m}{V} = 0$$.
or
$$ \rho_{0} (1 - \alpha h) -\rho_{0} (1 - \alpha h) \gamma (1 - \beta h) - \frac{m}{V} = 0 $$

Factor:

$$\rho_{0} (1 - \alpha h) ( 1 - \gamma (1 - \beta h) ) - \frac{m}{V} = 0 $$

Multiply both sides by ##V##:

$$\rho_{0} (1 - \alpha h) ( 1 - \gamma (1 - \beta h) )V - m = 0 $$

That is the same relationship I got! Multiply everything by ##g## and we are back at the working that used Newtons Second Law directly.

Yes, correct. The only question is whether the method is correct, but it seems to me that it is.

 

[erobz]

Yes, correct. The only question is whether the method is correct, but it seems to me that it is.

Where did all the stuff go about you were saying both signs are flopped ect... By deleting it you are making me look like a fool that is parroting everything you have already said for no reason...

Instead, simply say "I'm sorry, you are correct"

 

[Hak]

Where did all the stuff go about you were saying both signs are flopped ect... By deleting it you are making me look like a fool that is parroting everything you have already said for no reason...

I did not understand this statement. Sorry, I'm not a native English speaker...

 

[erobz]

I did not understand this statement. Sorry, I'm not a native English speaker...

You deleted all the stuff about disagreeing with the results being the same. Instead of just saying "yeah I goofed, sorry"

 

[Hak]

You deleted all the stuff about disagreeing with the results being the same. Instead of just saying "yeah I goofed, sorry"

Yes, because I realised after not even five seconds that I had made a mistake. Of course, I didn't delete it to avoid saying 'I was wrong, sorry', but only because I realised that your statement was right. I'm not a cocky guy, next time I won't delete, sorry...

 

[Hak]

May I ask what your final numerical result is?

 

[erobz]

Let's wait for someone else to verify. I'm currently baking a cake for my youngest daughters birthday, Then I have a wedding to attend.

 

[Hak]

Let's wait for someone else to verify. I'm currently baking a cake for my youngest daughters birthday, Then I have a wedding to attend.

Sorry, I couldn't have known. Here in Italy it's almost evening, I didn't notice the time difference. Happy birthday to your daughter, and have a wonderful evening at the wedding! Felicitations.

 

[Hak]

Do you all confirm this procedure and its numerical solution? Or is there some other more subtle demonstration? Thank you very much.

 

[Hak]

Do you all confirm this procedure and its numerical solution? Or is there some other more subtle demonstration? Thank you very much.

##h \approx 11 \ km## seems an exaggerated value, don't you think?

 

[erobz]

##h \approx 11 \ km## seems an exaggerated value, don't you think?

Not according to an internet search of maximum height reached by hot air balloon.

 

[Hak]

Not according to an internet search of maximum height reached by hot air balloon.

All right, thank you. Then I await confirmation, tell me if anyone is not convinced by this process and result. Thank you.

 

[erobz]

All right, thank you. Then I await confirmation, tell me if anyone is not convinced by this process and result. Thank you.

What is it that you want to investigate? Whether or the claims ##T = T_o(1 - \beta h )## and ##\rho = \rho_o( 1-\alpha h )## are realistic atmospheric models in the troposphere, or whether or not you've solved it correctly for the models given in the problem (because the latter has obviously been verified - at least for the process)?

 

[Hak]

What is it that you want to investigate? Whether or the claims ##T = T_o(1 - \beta h )## and ##\rho = \rho_o( 1-\alpha h )## are realistic atmospheric models in the troposphere, or whether or not you've solved it correctly for the models given in the problem (because the latter has obviously been verified - at least for the process)?

Both. Does the second option mean that the one above is the correct symbolic method to solve the problem?

 

[erobz]

Does the second option mean that the one above is the correct symbolic method to solve the problem?

Yes.

Both.

Then search "variation of temperature in troposphere vs altitude".

Then if you want to assume the ideal gas law model use that empirical result for the temperature to solve the hydrostatic differential equation for the density and compare with ##\rho = \rho_o( 1 - \alpha h)##

$$ \frac{dP}{dz} = -\rho g $$

 

[Hak]

Yes.

Then search "variation of temperature in troposphere vs altitude".

Then if you want to assume the ideal gas law model use that empirical result for the temperature to solve the hydrostatic differential equation for the density and compare with ##\rho = \rho_o( 1 - \alpha h)##

$$ \frac{dP}{dz} = -\rho g $$

OK, thank you. So, I have to substitute the expression of ##\rho## in the differential equation you reported below, and then what? What would be the empirical values to substitute?

 

[Hak]

OK, thank you. So, I have to substitute the expression of ##\rho## in the differential equation you reported below, and then what? What would be the empirical values to substitute?

This is what I found out.
https://seattlecentral.edu/qelp/set...ude because,above by incoming solar radiation.

 

[erobz]

OK, thank you. So, I have to substitute the expression of ##\rho## in the differential equation you reported below, and then what? What would be the empirical values to substitute?

I would substitute in for the Pressure ##P##, since the objective is to solve for ##\rho##.

 

[Hak]

I would substitute in for the Pressure ##P##, since the objective is to solve for ##\rho##.

OK, but what pressure value should I enter? I don't think I understand...

 

[erobz]

OK, but what pressure value should I enter? I don't think I understand...

You found that the variation is temperature in the troposphere is approximately linear. That agrees with ## T = ( T_o( 1- \alpha h ) ## (at least as a realistic model). Whether or not the slope is approximately correct in this problem you must compare it with a reputable source that states its value. In my fluids text book they state the temperature in the troposphere decreases linearly with increasing altitude at a lapse rate of ##5.87 \left[\frac{\text{K}}{ \text{km}}\right]##.

As for ##P## you don't enter a value, you enter the relationship( i.e. the model that relates the variables ##P##, ##\rho## and ##T## )

 

[Hak]

You found that the variation is temperature in the troposphere is approximately linear. That agrees with ## T = ( T_o( 1- \alpha h ) ## (at least as a realistic model). Whether or not the slope is approximately correct in this problem you must compare it with a reputable source that states its value. In my fluids text book they state the temperature in the troposphere decreases linearly with increasing altitude at a lapse rate of ##5.87 \left[\frac{\text{K}}{ \text{km}}\right]##.

As for ##P## you don't enter a value, you enter the relationship( i.e. the model that relates the variables ##P##, ##\rho## and ##T## )

This relationship would be ##P =\rho R T##?

 

[Hak]

This relationship would be ##P =\rho R T##?

Also, may I know what your fluids text book is called?

 

[erobz]

Also, may I know what your fluids text book is called?

Engineering Fluid Mechanics: Crowe,Elger,Williams,Roberson, Ninth Edition.

 

[Hak]

Yes.

Then search "variation of temperature in troposphere vs altitude".

Then if you want to assume the ideal gas law model use that empirical result for the temperature to solve the hydrostatic differential equation for the density and compare with ##\rho = \rho_o( 1 - \alpha h)##

$$ \frac{dP}{dz} = -\rho g $$

From this differential equation, I obtain:

$$\rho(h) = k e^{- \frac{gh}{RT}}$$.

What should I get, comparing with the expression of ##\rho## given in the text?