Bernoulli Problem (Somewhat Complicated?)

[atedinabox]

Homework Statement

So I'm dealing with a flow as seen in this image. There are 2 tanks, filled with 2 gases of different known densities, gas A and gas B.

Gas A flows from a tank with a static pressure sensor at point 1 through a line with diameter D₁.

Gas B flows from a tank with a static pressure sensor at point 3, through a pipe with diameter D₂ before being expanded at point 5 to also D₁.

The two flows mix at point 6, and then travel further down the line. There is a pressure sensor at point 7, and the flow continues onwards after that (no information known beyond this point).

Basically, all dimensions of the system are known (including the ones not pictured such as the pipe lengths) along with the static pressure sensors at points 1, 3, and 7, and the fluid densities, but the problem is to find out the mass flows. Considering all major and minor losses. Although the effect of different heights can be neglected!

I'm not exactly sure how to deal with the 2 gases of different densities, and it's made even more confusing for me with the rightmost pipe having varying velocities and that impacting the losses. Any tips would be greatly appreciated! And I know how to account for the major/minor losses already (meaning the loss factors associated with the pipe roughness/bends/expansion/merge/etc) so I'm okay there. But not where the velocities come into play as part of those losses.

Homework Equations

Apply Bernoulli (many times!) to in and outlets:
.5*ρ*V₁² + P₁ - ΔP= .5*ρ*V₂² + P₂

where ΔP is .5*ρ*V²*(f*L/D+ƩK)

with K being the minor loss coefficient.

The Attempt at a Solution

To be honest I'm not even sure where to start. Is the velocity in the two tanks zero, or can it be considered such?

For left tank:
P₁ - ΔP (from the tank exit) = P₂ + .5*ρ_A*V₂²

For right tank:
P₃ - ΔP = P₄ + .5*ρ_B*V₄²

V₄*D₂ = V₅*D₁

P₄ + .5*ρ_B*V₄² - ΔP (from expansion and friction) = P₅ + .5*ρ_B*V₅²

I'm not really too sure about any of this let alone what comes next! ... Thanks so much for any help!

 

[KataKoniK]

Thank you for your post. I understand your confusion with the two gases of different densities and the varying velocities in the rightmost pipe. To start, the velocities in the two tanks can be considered negligible, as the gases are in a static state before they start flowing.

For the left tank, you are correct in your equation: P1 - ΔP (from the tank exit) = P2 + .5*ρA*V22. This takes into account the pressure drop due to friction and minor losses as the gas flows through the pipe.

For the right tank, you will need to take into account the change in density as the gas expands at point 5. This can be done using the ideal gas law: P1V1/T1 = P2V2/T2. Since the temperature is constant, we can simplify this to P1V1 = P2V2, where P1 and V1 are the pressure and volume at point 4, and P2 and V2 are the pressure and volume at point 5.

Next, you will need to take into account the change in velocity as the gas expands. This can be done using the continuity equation: A1V1 = A2V2, where A1 and V1 are the cross-sectional area and velocity at point 4, and A2 and V2 are the cross-sectional area and velocity at point 5.

Now, with the ideal gas law and continuity equation, you can solve for the velocity at point 5 and use it in your equation: P4 + .5*ρB*V42 - ΔP (from expansion and friction) = P5 + .5*ρB*V52. This will give you the pressure at point 5, which you can then use to calculate the pressure drop and velocity at point 6 where the two gases mix.

I hope this helps. Let me know if you have any further questions or if you need more clarification. Good luck with your calculations!