Which Equations Solve Fluid Dynamics in a Pressurized Container?

[Michael K]

I am trying to solve for the temperature/pressure profiles of a small closed-end (one side) tube inside a pressurized container kept at constant temp./pressure. For now, I am assuming that vacuum conditions exist within the tube. I am not sure, however, which equations I should be using.

Initially I reasoned as follows: For molecules just entering the tube opening (at t=0) I would have to use effusion rates to calculate P(t) existing at the opening; and then, afterwards, plug this into Poiseuille's equation to get a picture of overall pressure buildup over time. Is this correct? Do I need to work with effusion rates, or, is there some other, more correct, (or, better) way to solve this?

Also, I am aware that the effusion equation only applies to conditions where the area of the hole is smaller in scale to the mean free paths of gaseous molecules. The scenario I am working with calls for a tube diameter within mm range (length is ~ meter range) at pressures above atm (2-3) and temperatures above 100 C. I believe that the equation no longer apllies. Is this correct? Are there any alternatives?

Finally, I have solved the differential eqn. for effusion into a vacuum, but I believe that my answer is incorrect because I assumed temperature remains constant. Can I make this assumption? If not, how do I rectify my diff. eqn? ( dP/dt = kT/V dN/dt, where dN/dt =
(Po -P(t))A / sqrt(2piMRT) Po = pressure of container, k = Boltzman's constant M = molecular mass, A = area)

Any help would be appreciated.

 

[LightHero]

The correct approach for solving the temperature/pressure profiles in a small closed-end tube inside a pressurized container kept at constant temperature and pressure is to first calculate the effusion rate of gas molecules entering the tube. This can be done using the effusion equation, which states that the effusion rate is proportional to the ratio of the pressure of the container to the pressure of the tube, as well as the area of the hole. Once you have the effusion rate, then you can use Poiseuille's equation to calculate the pressure buildup over time. Additionally, you can then use the ideal gas law to calculate the temperature of the gas inside the tube. Regarding your assumption of constant temperature, this assumption is not correct. The temperature of the gas inside the tube will depend on the pressure of the container, as well as the pressure of the tube, so it will not remain constant. Therefore, you will need to solve the differential equation for pressure (dP/dt = kT/V dN/dt) while incorporating the changing temperature of the gas.

 

[astrokreng]

I would first like to compliment you on your clear and detailed explanation of your problem. It shows that you have put a lot of thought and effort into understanding the system you are studying.

To answer your question, there are a few different equations that could potentially be used to solve for the temperature and pressure profiles in your closed-end tube scenario. One approach could be to use the ideal gas law, which relates pressure, volume, and temperature for a gas. However, this may not be applicable in your case since you mention that the vacuum conditions within the tube may not be suitable for this equation.

Another approach could be to use the Navier-Stokes equations, which are commonly used in fluid dynamics to describe the motion of fluids. These equations take into account factors such as viscosity and density, which may be important in your scenario.

In terms of your question about using effusion rates, this may not be the best approach since effusion is typically used to describe gas flow through small openings, and your tube diameter is in the millimeter range. Additionally, as you mentioned, the effusion equation may not apply at the pressures and temperatures you are working with.

Regarding your concern about assuming constant temperature, it is important to consider the conditions within your closed-end tube and how they may change over time. If the temperature within the tube is not constant, then your assumption may not be accurate and could lead to incorrect results. In this case, it may be necessary to include a temperature term in your differential equation to account for any changes in temperature.

Overall, it is important to carefully consider the specific conditions and variables in your system and choose the appropriate equations and models to accurately describe and solve for the temperature and pressure profiles. It may also be helpful to consult with other experts in the field of fluid dynamics for further guidance and suggestions.