Modeling boiling in a closed container with a small hole

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Discussion Overview

The discussion revolves around modeling the boiling process in a closed container with a small hole, specifically in the context of a system similar to Heron's Aeolipile. Participants explore the relationships between heat flux, pressure, temperature, and mass flow in the system.

Discussion Character

  • Exploratory
  • Technical explanation
  • Mathematical reasoning

Main Points Raised

  • One participant seeks to determine the pressure and temperature in the container based on a known heat flux, assuming that boiling will stabilize the pressure and temperature of the vapors.
  • Another participant suggests that the exit velocity of the gas relates to the pressure difference, indicating that equilibrium occurs when enough heat is transferred to maintain mass flow, with temperature expected to align with vapor pressure at standard conditions.
  • A question is raised about the expected pressure increase and steam mass outflow once steady-state boiling is achieved, along with a request for calculation methods.
  • It is noted that the rate of heat flow to the container is crucial, as it influences all other parameters in the model.
  • Assuming pool boiling in the nucleate regime, one participant points out that the heat flux equation involves both saturation and surface temperatures, complicating the model due to the interdependence of these variables.
  • Another participant mentions that if the assumption of all water being at 100°C is insufficient, a more detailed model must account for heat and gas flow, highlighting the balance between added details and the number of unknowns.
  • A participant shares their past experience deriving a closed-form solution for pressure and mass flow rates in a similar setup, encouraging others to attempt modeling the system.

Areas of Agreement / Disagreement

Participants express various viewpoints and uncertainties regarding the modeling approach, the relationships between parameters, and the assumptions necessary for accurate calculations. No consensus is reached on the best method or specific outcomes.

Contextual Notes

Participants acknowledge the complexity of the model due to the interdependence of variables such as heat flux, saturation temperature, and surface temperature, as well as the potential for multiple unknowns arising from added details.

Hang11
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I'm trying to build a mathematical model of something like Heron's Aeolipile:
http://en.wikipedia.org/wiki/Aeolipile
I'd like to know, based on a known heat flux, the pressure and temperature attained in the container.
I assume as water boils, the control volume loses mass and energy, the pressure and temperature of the vapors will stabilize.
 
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The exit velocity of the gas (and therefore mass flow) will have some relation to the pressure difference. Equilibrium happens where you can boil enough water (transfer enough heat) to maintain that mass flow. Temperature is given by the vapour pressure - I would expect no significant deviations from standard pressure and 100°C.
 
but what kind of pressure increase in the vessel and what steam mass outflow should I get once it reaches a steady state boil?
How would I calculate that?
 
You'll need the rate of heat flow to the container, everything else will follow from that. That rate will depend on your heating mechanism.
 
Assuming pool boiling, nucleate regime, the equation of the heat flux contains both the saturation temperature and the surface temperature. However, as more heat is pumped into the system (the exhaust is small), the Tsat changes. So , even knowing the heat flux, I still have two unknowns, the surface temperature and the saturation temperature.
 
If the approximation "all water is at 100°C" is not precise enough for your model then you'll have to model heat and gas flow in the water. Every new detail you add in the model gives the same number of unknown quantities as it gives equations to calculate those, as long as you include all relevant material properties and so on.
 
40 years ago, I derived a closed form solution for dP/dt, dm(water)/dt, and dm(steam)/dt in a setup very similar to this. It was great fun deriving it from mass balances of water and steam, and energy balances of water and steam, and the properties of water and steam. I used the result in simulators for nuclear power plants.

Give it a try. You'll succeed if you persevere.
 

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