Compression power equation of a gas mixture

[streamlinerr]

Homework Statement

I have a mixture of hydrogen (H₂) and water (vapor) that is compressed from pressure p₁ to p₂. Mass flow of the mixture is known, as well as mass fraction of the hydrogen in the mixture.

Known data:
Pressure p₁
Pressure p₂
Temperature T
mass flow Mf
mass fraction w_H2

I need to develop an equation for the power of a compressor as a function of the above data.

Homework Equations

Adiabatic compressor power equation

Equation of state in the form of:
p = ρ * R * T

with ρ being the density and R a specific gas constant

The Attempt at a Solution

At first I thought this should be easy, however I stumbled on a problem pretty quickly.

In the adiabatic compressor power equation we have a volume flow Q, which is a mass flow Mf divided by density ρ. I tried developing an equation for the density of a mixture from an equation of state:

ρ = p/T * 1/R

Since mixture density is a combination of both densities i thought to calculate the mixture density the following way:
ρ_mixt = ρ_H2 * w_H2 + ρ_H2O * (1-w_H2) = p/T * ( w_H2 / R_H2 + (1-w_H2) / R_H2O)

with R_H2 and R_H2O being the specific gas constants of hydrogen and water vapor.

I'm not sure if this the correct way to calculate the mixture density. Is the use mass fraction correct or should i use volume fraction?

Second problem arises when trying to determine heat capacity ratio γ that appears in the compression equation. I've been looking how to determine γ of a mixture (as a function of properties given above), but failed so far.

I'd appreciate some help on this issue.

Thanks.

 

[KataKoniK]

Thank you for your question. It seems like you are on the right track in your attempt to develop an equation for the power of a compressor as a function of the given data. However, there are a few things that need to be clarified in order to arrive at a correct solution.

Firstly, the equation of state that you have used, p = ρ * R * T, is only valid for an ideal gas. In this case, both hydrogen and water vapor can be considered as ideal gases, but the mixture of the two will not behave as an ideal gas. Therefore, you will need to use a more complex equation of state, such as the Peng-Robinson equation, to accurately calculate the density of the mixture. This equation takes into account the interactions between the molecules of the two gases in the mixture.

Secondly, when calculating the density of the mixture, you will need to use the volume fraction instead of the mass fraction. This is because the density of a mixture is the sum of the densities of its components, weighted by their respective volume fractions.

Lastly, the heat capacity ratio γ is also a function of the composition of the mixture. You can use the law of corresponding states to calculate the mixture's heat capacity ratio, which takes into account the composition and temperature of the mixture.

I hope this helps you in developing an accurate equation for the power of the compressor. If you have any further questions, please don't hesitate to ask. Good luck!