Is my understanding of the hot air balloon equation correct?

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Discussion Overview

The discussion revolves around the understanding and application of the hot air balloon equation, specifically addressing the implications of derived temperatures in relation to lift and buoyancy. Participants explore the theoretical aspects of the equation and its physical interpretations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant presents an equation for calculating the temperature of air in a hot air balloon based on load, pressure, volume, and ambient temperature, expressing confusion over negative temperature results.
  • Another participant emphasizes the importance of specifying units for the variables in the equation to ensure dimensional consistency.
  • A third participant clarifies that the derived equation can yield negative temperatures under certain conditions, specifically when the load exceeds the buoyant force provided by the displaced air, suggesting that this scenario is theoretically valid but physically unreasonable.
  • The ideal gas law is referenced, indicating that negative temperatures imply negative mass, which raises questions about the physical feasibility of such conditions.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the correctness of the equation or the understanding of its implications, as there are differing views on the physical interpretation of negative temperatures and their relevance to the problem at hand.

Contextual Notes

There are limitations regarding the assumptions made about the variables involved, particularly concerning the relationship between load, temperature, and buoyancy. The discussion does not resolve the mathematical implications of negative temperatures in the context of the ideal gas law.

Poseidonho
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I had meet a problem when I study the concept of the hot air balloon. The equation was giving
Load = P*V*(1/TempAir - 1/TempBalloon)/R

Then I reverse to

TempBalloon = 1/[1/TempAir -Load*R/(P*V)]

When the volume is less then 1m^3, the temperature calculated giving negative temperature (In kelvin and degree also) but suppose had a greater in temperature with my understanding for the theory.

So is that any wrong in the equation or my understanding?
 
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In order for the units of your formula to balance correctly, the following quantities must be specified:

Load in kg

Temperatures are in degrees Kelvin

Pressure in Pascals

Volume in cubic meters

R for air is 287.058 J/kg/K
 
Poseidonho said:
So is that any wrong in the equation or my understanding?

As I understand it, you derived an equation that gives the required temperature to give a specified lift, holding everything else (ambient temperature, ambient pressure, volume of the balloon) constant. Now you wonder -- can the equation be correct even though it sometimes gives a negative temperature as a result?

Yes, it can be correct. If the load you are trying to lift is heavier than the mass of the air displaced by the balloon then no matter how hot you heat the air in your balloon, it can never be light enough to have a net upward buoyancy greater than the weight of the load.

The ideal gas law says that a fixed volume with a negative temperature has a negative mass. This is not physically reasonable, but does account for the solution you are getting when the load is too high.
 
Thank You very much, I understand d
 

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