Is Exponential Form Better for Balloon with Inserted Load Calculation?

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The discussion focuses on calculating the buoyant force acting on a balloon with inserted load, using Archimedes' principle and the ideal gas law. The participants explore the relationship between the densities of the air inside and outside the balloon, emphasizing the importance of temperature and pressure variations with altitude. A key point raised is the need to correctly account for the buoyant force and the average density of the balloon system, which includes the air inside, the envelope, and the load. There is a consensus that simplifying assumptions can lead to significant errors in height calculations, and the correct interpretation of the variables involved is crucial. Ultimately, the conversation highlights the complexities of buoyancy calculations in variable atmospheric conditions.
  • #101
Hak said:
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## )
 
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  • #102
erobz said:
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##?
 
  • #103
Hak said:
This relationship would be ##P =\rho R T##?
Also, may I know what your fluids text book is called?
 
  • #104
Hak said:
Also, may I know what your fluids text book is called?
Engineering Fluid Mechanics: Crowe,Elger,Williams,Roberson, Ninth Edition.
 
  • #105
erobz said:
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?
 
  • #106
Hak said:
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?
How about you show some steps?
 
  • #107
erobz said:
How about you show some steps?
Yes, I show the steps now, but is this result right?
 
  • #108
Hak said:
Yes, I show the steps now, but is this result right?
I don't know, It's not an explicit result in my book. I would like to see the steps you have taken to solve the equation. If each main step was executed correctly, then the result would be too.
 
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  • #109
erobz said:
I don't know, It's not an explicit result in my book. I would like to see the steps you have taken to solve the equation. If each main step was executed correctly, then the result would be too.
OK, I will insert all the steps shortly. In case my result is right, what should I get from comparing the two expressions? In other words, what considerations should I make?
 
  • #110
Hak said:
OK, I will insert all the steps shortly. In case my result is right, what should I get from comparing the two expressions? In other words, what considerations should I make?
One step at a time please. If you show some basic steps how you arrived at the solution ( verifying if it is correct or not- something you are very persistent in having others do in your problems), we can move forward with what it means.
 
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  • #111
erobz said:
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)##

$$ \frac{dP}{dz} = -\rho g $$
Preface: I am very insistent that others check my work because this is the only way to understand whether my theories and procedures are reasonable or not. I hope I have not annoyed anyone.

Below are the steps I took to calculate the differential equation ## \frac{dP}{dh} = -\rho g \Rightarrow \ RT \frac{d \rho}{dh} = -\rho g##.

This equation can be written as:

$$\rho' + \frac{g}{RT} \rho = 0$$.

It is a first-order linear ordinary differential equation of the type:

$$y'(h) + a_0 (h) y(h) = 0$$, which has solution ## y(h) = k \cdot e^{-A(h)}##, with ##A(h) = \int a_0(h) dh##.

Since, in this case, ##A(h) = \frac{g}{RT} h##, we have:

$$\rho (h) = k \cdot e ^{- \frac{g}{RT} h}$$

What do you think?
 
  • #112
Hak said:
Below are the steps I took to calculate the differential equation ## \frac{dP}{dh} = -\rho g \Rightarrow \ RT \frac{d \rho}{dh} = -\rho g##.
##T## isn't a constant, you can't take it out of the derivative.

$$ \frac{d}{dh} ( \rho T ) = - \frac{\rho g }{R} $$

Continue from here.
 
  • #113
erobz said:
##T## isn't a constant, you can't take it out of the derivative.

$$ \frac{d}{dh} ( \rho T ) = - \frac{\rho g }{R} $$

Continue from here.
You are right. I'm having difficulty solving this ODE, I can't figure out how to work with another dependent variable from ##h##. Do you have any advice?
 
  • #114
Hak said:
You are right. I'm having difficulty solving this ODE, I can't figure out how to work with another dependent variable from ##h##. Do you have any advice?
##T## is a function of ##h##, use the product rule to complete the derivative on the LHS.
 
  • #115
erobz said:
##T## is a function of ##h##, use the product rule to complete the derivative on the LHS.
Following your advice, I arrive at the equation:

$$\rho' T + T' \rho+ \frac{g}{R} \rho = 0 $$, from which:

$$\rho' T + T_0 \beta \rho + \frac{g}{R} \rho = 0 $$. Therefore:

$$\rho' T + \left(T_0 \beta + \frac{g}{R} \right) \rho = 0$$, whence:

$$\rho' + \frac{1}{T} \left(T_0 \beta + \frac{g}{R} \right) \rho = 0$$.

Following the procedure of the previous message, we have:

$$A(h) = \int \frac{1}{T(h)} \left(T_0 \beta + \frac{g}{R} \right) dh = \left(T_0 \beta + \frac{g}{R} \right) \int \frac{1}{T_0(1 + \beta h)} dh = \left(T_0 \beta + \frac{g}{R} \right) \frac{ln |1 + \beta h|}{T_0 \beta} h + c$$. Finally:

$$\rho (h) = k \cdot e ^{-\left(T_0 \beta + \frac{g}{R} \right) \frac{ln |1 + \beta h|}{T_0 \beta } h + c}$$.

What do you think?
 
  • #116
There is a sign error here, note that ##T = T_o - T_o \beta h ##.
Hak said:
$$\rho' T + T_0 \beta \rho + \frac{g}{R} \rho = 0 $$.

Hak said:
$$\rho' + \frac{1}{T} \left(T_0 \beta + \frac{g}{R} \right) \rho = 0$$.
Once you fix the sign error write

$$ \frac{d \rho}{dh} = \cdots $$

Then separate variables and integrate both sides.
 
  • #117
Ok, I'm fixing the sign error.$$\rho' T + T' \rho + \frac{g}{R} \rho = 0 $$, from which:
$$\rho' T - T_0 \beta \rho + \frac{g}{R} \rho = 0 $$. Therefore:
$$\rho' T + \left(\frac{g}{R} - T_0 \beta \right) \rho = 0$$, whence:
$$\rho' + \frac{1}{T} \left(\frac{g}{R} - T_0 \beta \right) \rho = 0$$.
Following the procedure of the previous message, we have:
$$A(h) = \int \frac{1}{T(h)} \left(\frac{g}{R} - T_0 \beta \right) dh = \left(\frac{g}{R} - T_0 \beta \right) \int \frac{1}{T_0(1 - \beta h)} dh = - \left(\frac{g}{R} - T_0 \beta \right) \frac{ln |1 - \beta h|}{T_0 \beta} h + c$$. Finally:
$$\rho (h) = k \cdot e ^{\left(\frac{g}{R} - T_0 \beta \right) \frac{ln |1 - \beta h|}{T_0 \beta } h + c}$$.
Is it really necessary to trace the ODE back to a separable variable differential equation? Is my procedure wrong? I can't do it directly, however...
 
  • #118
Hak said:
Is it really necessary to trace the ODE back to a separable variable differential equation? Is my procedure wrong?
Yes. You are not getting the proper result.
 
  • #119
erobz said:
Yes. You are not getting the proper result.
Okay. I get to the equation:

$$T \ d \rho = \left(T_0 B - \frac{g}{R} \right) \rho (h) dh$$

I don't know how to handle this equation by integrating both members....
 
  • #120
Hak said:
Okay. I get to the equation:

$$T \ d \rho = \left(T_0 B - \frac{g}{R} \right) \rho (h) dh$$

I don't know how to handle this equation by integrating both members....
The method of separation of variables... All factors involving ##\rho## on the left, and everything involving ##h## on the right.

$$ \frac{d \rho}{\rho} = \left(T_0 \beta - \frac{g}{R} \right) \frac{dh}{T(h)} $$
 
  • #121
erobz said:
The method of separation of variables... All factors involving ##\rho## on the left, and everything involving ##h## on the right.

$$ \frac{d \rho}{\rho} = \left(T_0 \beta - \frac{g}{R} \right) \frac{dh}{T(h)} $$
Thanks. I get:

$$ ln |\rho| + c_1 = - \left(T_0 \beta - \frac{g}{R} \right) \frac{ln |1 - \beta h|}{T_0 \beta} + c_2$$, from which:

$$ \rho (h) = e^{- \left(T_0 \beta - \frac{g}{R} \right) \frac{ln |1 - \beta h|}{T_0 \beta} + c}$$.

This is the same result as before, with the difference that I had previously inserted an extra ##h## term, which was wrong.
 
  • #122
Hak said:
Thanks. I get:

$$ ln |\rho| + c_1 = - \left(T_0 \beta - \frac{g}{R} \right) \frac{ln |1 - \beta h|}{T_0 \beta} + c_2$$, from which:

$$ \rho (h) = e^{- \left(T_0 \beta - \frac{g}{R} \right) \frac{ln |1 - \beta h|}{T_0 \beta} + c}$$.

This is the same result as before, with the difference that I had previously inserted an extra ##h## term, which was wrong.

It may be the same, but you are overcomplicating it and making it virtually unrecognizable. Use definite integrals on each side:

$$ \int_{\rho_o}^{\rho(h)} \frac{d \rho}{\rho} =\left(T_0 \beta - \frac{g}{R} \right)
\int_0^h \frac{dh}{T_o - T_o \beta h } $$
 
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  • #123
erobz said:
It may be the same, but you are overcomplicating it and making it virtually unrecognizable. Use definite integrals on each side:

$$ \int_{p_o}^{\rho(z)} \frac{d \rho}{\rho} =\left(T_0 \beta - \frac{g}{R} \right)
\int_0^h \frac{dh}{T_o - T_o \beta h } $$
OK. I get:
$$ln |\rho (z)| - ln|\rho_0| = \frac{ln |T_0| - ln |T_0 (1- \beta h)|}{T_0 \beta} \left(T_0 \beta - \frac{g}{R} \right) $$. Right?

Edit. I had forgotten part of the equation.
 
  • #124
Hak said:
OK. I get:
$$ln |\rho (z)| - ln|\rho_0| = \frac{ln |T_0| - ln |T_0 (1- \beta h)|}{T_0 \beta}$$. Right?
Close, but you are forgetting the factor out front, and use log rules to turn differences of logs into a log of a quotient, and then rules about logs and factors, and exponents etc... once complete, exponentiate both sides.
 
  • #125
erobz said:
Close, but you are forgetting the factor out front, and use log rules to turn differences into quotients, and logs and factors.
OK. I get:

$$ln |\frac{\rho(z)}{\rho_0}| = - ln |1 - T_0 \beta| \left(1 - \frac{g}{T_0 \beta R} \right) $$, from which:

$$\rho (z) = \rho_0 \cdot e ^ {- ln |1 - T_0 \beta| \left(1 - \frac{g}{T_0 \beta R} \right)}$$.
Right?
 
  • #126
Hak said:
OK. I get:

$$ln |\frac{\rho(z)}{\rho_0}| = - ln |1 - T_0 \beta| \left(1 - \frac{g}{T_0 \beta R} \right) $$, from which:
Try again. The difference of logs on the RHS, where did it go?

Then...Rules about logs: specifically ##a ln(x) = ln(x^a)##

Only then do you exponentiate both sides.
 
  • #127
erobz said:
Try again. The difference of logs on the RHS, where did it go?
It is correct, or not? We have:

$$ln |T_0| - ln |T_0 (1 - \beta h)| \Rightarrow \ ln |\frac{T_0}{T_0 (1 - \beta h)}| = ln |\frac{1}{(1 - \beta h)}| = ln |(1 - \beta h)|^{-1} = - ln |(1 - \beta h)| $$.

Where is the mistake?
 
  • #128
Hak said:
It is correct, or not? We have:

$$ln |T_0| - ln |T_0 (1 - \beta h)| \Rightarrow \ ln |\frac{T_0}{T_0 (1 - \beta h)}| = ln |\frac{1}{(1 - \beta h)}| = ln |(1 - \beta h)|^{-1} = - ln |(1 - \beta h)| $$.

Where is the mistake?
Sorry, Thats fine. I missed that you factored out ##T_o##.

But also apply log power rule to the factor out front to bring it inside the logarithm as an exponent. You want to be ending with this form, before exponentiation:

$$\ln (a) = \ln(b^c) $$
 
  • #129
erobz said:
But also apply log power rule to the factor out front to bring it inside the logarithm
OK, I get:

$$ln |(1- \beta h) ^ {(1 - \frac{g}{T_0 \beta R})}|$$.

Right?
 
  • #130
Hak said:
OK, I get:

$$ln |(1- T_0 \beta) ^ {(1 - \frac{g}{T_0 \beta R})}|$$.

Right?
Yeah, for the right hand side. Then exponentiate both sides.
 
  • #131
I get:
$$\rho (z) = \rho_0 |(1- \beta h) ^ {(1 - \frac{g}{T_0 \beta R})}|$$.

Right?
 
  • #132
Hak said:
I get:
$$\rho (z) = \rho_0 |(1- \beta h) ^ {(1 - \frac{g}{T_0 \beta R})}|$$.

Right?
I get the negative of that exponent. ## \left( \frac{g}{RT_o \beta} - 1 \right)##
 
  • #133
erobz said:
I get the negative of that exponent. ## \frac{g}{RT_o \beta - 1}##
Yes, you are right, I forgot to include it.
 
  • #134
$$\rho (z) = \rho_0 |(1- \beta h) ^ {(\frac{g}{T_0 \beta R} - 1)}|$$ is the right expression.
 
  • #135
Now, what should be done?
 
  • #136
Hak said:
$$\rho (z) = \rho_0 |(1- \beta h) ^ {(\frac{g}{T_0 \beta R} - 1)}|$$ is the right expression.
you don't need the absolute values
 
  • #137
erobz said:
you don't need the absolute values
OK, thanks. Now?
 
  • #138
Hak said:
OK, thanks. Now?
Now, it should be clear that the function given for ##\rho## in the problem statement does not strictly adhere to the ideal gas law assumption that is used to solve the problem. Using ideal gas law, we see a power law. In the problem it was linear.
 
  • #139
erobz said:
Now, it should be clear that the function given for ##\rho## in the problem statement does not strictly adhere to the ideal gas law assumption that is used to solve the problem. Using ideal gas law, we see a power law. In the problem it was linear.
Right, thank you. What conclusions can be drawn from this?
 
  • #140
Hak said:
Right, thank you. What conclusions can be drawn from this?
I don't know, but it could be that the linear function approximates the power law reasonably well over these altitudes, or they simply chose a function for ##\rho## which makes the problem solvable algebraically. Yet another interpretation is that the linear function is empirical, and its the ideal gas law that was the issue. You have to plot the functions to see why they might have done what they have done.
 
  • #141
erobz said:
I don't know, but it could be that the linear function approximates the power law reasonably well over these altitudes, or they simply chose a function for ##\rho## which makes the problem solvable algebraically. You have to plot the functions to see why they might have done what they have done.
I will try to trace them. However, I am noticing that in the expression of the problem a term ##\alpha## pops up, whereas here there is only one term in ##\beta##. What does this mean?
 
  • #142
The only real difference between the two functions seems to be that the function given by the text has an unlimited domain, whereas the one we have just calculated has an upper limit, so it has a limited domain. There does not seem to be any substantial difference.
 
  • #143
Hak said:
I will try to trace them. However, I am noticing that in the expression of the problem a term ##\alpha## pops up, whereas here there is only one term in ##\beta##. What does this mean?
##\alpha## either characterizes the linear function they made up for ##\rho ~\text{vs.}~ h## or the linear function they observe for ##\rho~ \text{vs.}~ h##, just as ##\beta## characterizes the approximately linear function they observe for ##T## vs. ##h## in the troposphere. You have to do some calculation to see what is what. I can't tell you anymore, because I'm not doing all that.
 
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  • #144
Hak said:
The only real difference between the two functions seems to be that the function given by the text has an unlimited domain, whereas the one we have just calculated has an upper limit, so it has a limited domain. There does not seem to be any substantial difference.
What do you think?
 
  • #145
Hak said:
What do you think?
Then I think you have solved the mystery. The linear function approximates the power law ( strict ideal gas law treatment ) very well over the domain in question, and (this is the important part) ...it also makes the problem algebraically solvable. Or it too is an empirical result ( i.e. the atmosphere isn't really quite ideal ) like the temperature function, and it makes the problem solvable. It doesn't matter either way.
 
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  • #146
erobz said:
Then I think you have solved the mystery. The linear function approximates the power law ( strict ideal gas law treatment ) very well over the domain in question, and (this is the important part) ...it also makes the problem algebraically solvable. Or it too is an empirical result like the temperature function, and it makes the problem solvable. It doesn't matter either way.
OK, thank you, @erobz. You gave me a huge amount of help.
 
  • #147
This is not rocket science.....it is balloon science. As the balloon ascends into cooler air that effect will increase the lift of the 100 deg (differential??) T this is not stated clearly at all. The air pressure is of course exponential with z so this prblem is sort of silly. Assume a uniform temperature and the simple exponential atmosphere and try for a good solution. This is a simple exercise, unless one chooses to make it otherwise. Yikes.
 
  • #148
hutchphd said:
This is not rocket science.....it is balloon science. As the balloon ascends into cooler air that effect will increase the lift of the 100 deg (differential??) T this is not stated clearly at all.
I struggle to understand this statement. Could you explain it again? Thank you very much.
 
  • #149
hutchphd said:
The air pressure is of course exponential with z so this prblem is sort of silly. Assume a uniform temperature and the simple exponential atmosphere and try for a good solution. This is a simple exercise, unless one chooses to make it otherwise. Yikes.
Excuse me, but I cannot understand what the crux of what you want to say is. Could you explain that too? Thank you again.
 
  • #150
$$\rho(z,T)=\rho(z=0,T)e^{-\alpha z}$$and from the Ideal gas equation $$F_{lift}=\rho gV\frac{\Delta T} T$$the buoyancy is the mass of the air expelled because the ballon is heated. Also as mentioned N.B. $$\alpha\approx1/(20km)$$
REVISED
 
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