Phase equilibrium in a vessel with unsteady flow

[fizzybiz]

Hi all - my first post. What a great resource!

Homework Statement

I have an initially closed, pressurised vessel at ambient temperature, with a single species in it at dynamic equilibrium with some volume fraction of vapour and liquid. I am in the "wet vapour" region of the P-v-T surface, described by the Clausius-Clapeyron equation.

I then draw some mass of liquid out. I view the problem in discreet terms, but could possibly model with a mass flow rate out.

In the real world, the vessel will cool - energy will be drawn into the vapour - partially from the effect of vapour expansion (thinking of the liquid rather like a piston in a cylinder) and partially from the effect of vapourisation of some of the remaining liquid. I presume if the mass flow rate is low, the expansion effect will be negligible. I want to quantify the energy flux due to vapourisation.

Homework Equations

Clausius-Clapeyron equation, describing vapour pressure at some temperature - the vapour-liquid equilibrium line in the P-T plane. ln(p2/p2) = (dHvap/R)(1/T2 - 1/T1)

Ideal gas law; pV = mRT

(heat flux at constant volume; Q = m.cv.dT; not used)

Possibly use the relative volatility theory described here (http://tinyurl.com/676gsb) for a binary, 2-species problem.

The Attempt at a Solution

I have approached this by first presuming the temperature recovers to ambient at every liquid mass increment drawn from the vessel. I have no model for heat flux across the vessel wall.

I draw mass m of liquid from the tank, then the volume change of vapour is easily calculated from mass/liquid density. I presume if the temperature remains constant that we equilibrate back along the clausius-clapeyron line, and that the vapour pressure remains constant also (not sure if that's valid). Then the new mass of vapour can be found from pV = mRT. The energy flux drawn into vapourise the mass difference is calculated from Q = dHvap.m.

I'd like to know the volume fractions of vapour and liquid, but I'm not sure if that's useful/misleading.

 

[Nate810]

I'm also not sure if this approach is valid. The main challenge I think is modelling the cooling effect of the incoming mass flow. I'm happy to hear any thoughts on this.Thanks in advance!Nick

 

[Moetasim]

Phase equilibrium in a vessel with unsteady flow is a complex problem that requires careful consideration of thermodynamic principles. The Clausius-Clapeyron equation is a useful tool for determining the vapor pressure at a given temperature, but it is important to keep in mind that this equation assumes a constant temperature and does not account for the effects of energy transfer during the phase change. In order to accurately model the system, it may be necessary to incorporate heat transfer equations and consider the effects of temperature changes on the vapor-liquid equilibrium line.

In regards to the energy flux due to vaporization, it is important to consider the specific conditions of the system, such as the rate of mass flow and the heat transfer across the vessel wall. The ideal gas law can provide a good estimate of the volume change of the vapor, but it may not accurately account for the effects of non-ideal behavior at high pressures. Additionally, the relative volatility theory may be helpful for a binary, two-species problem, but it may not accurately represent the behavior of a single species system.

Overall, it is important to carefully consider all aspects of the system and to use appropriate equations and models to accurately describe the phase equilibrium in a vessel with unsteady flow.

 

[baba89]

Hi there, it sounds like you are studying phase equilibrium in a vessel with unsteady flow. This is a complex and interesting topic in thermodynamics! The Clausius-Clapeyron equation is definitely a useful tool for describing the relationship between temperature and pressure at the interface between liquid and vapor phases. It is important to note, however, that this equation assumes a system at equilibrium and does not take into account any changes in temperature or pressure due to flow.

In your scenario, where you are drawing mass out of the vessel, it is likely that the temperature and pressure will change as the system tries to reach a new equilibrium. The energy flux due to vaporization can be calculated using the Clausius-Clapeyron equation, but it may be more accurate to use a more comprehensive model that takes into account the dynamic changes in temperature and pressure.

The relative volatility theory you mentioned could be useful in a binary system, but it may be more challenging to apply in a single species system. It may be worth considering other thermodynamic models, such as the Peng-Robinson equation, which can account for non-ideal behavior and changes in temperature and pressure.

Additionally, in order to accurately quantify the energy flux due to vaporization, it would be important to consider heat transfer across the vessel walls. This can be a complex problem, but there are various models and equations that can be used to estimate this heat transfer.

Overall, the volume fractions of vapor and liquid can be useful in understanding the behavior of the system, but it is important to also consider the changes in temperature and pressure due to unsteady flow. I hope this helps and good luck with your studies!