Energy conservation in mixing streams

[pawel]

Is it correct to apply two separate Bernoullis equation - for sections 1-3 and for 2-3? Why/why not?

I ve seen such solution to this problem on the internet but I have some doubts.
I think that energy between sections 1 and 3 is not conserved because of energy exchange with the other stream.

Energy conservation for the whole system
Q1(v1^2/2 +p1/rho)+Q2(v2^2/2 +p2/rho)=(Q1+Q2)(v3^2/2 +p3/rho)
is not enough to determine (p3-p2) needed for Y momentum conservation.

system of:
v1^2/2 +p1/rho +deltaE=v3^2/2 +p3/rho
v1^2/2 +p1/rho -deltaE=v3^2/2 +p3/rho
needs an additional equation as well

I would be thankful for every help

 

[benit13]

You also need to consider the continuity equation:

\(\displaystyle Q_3 = Q_1 + Q_2\)

\(\displaystyle Q_3 = A_1 v_1 + A_2 v_2\)

Since each pipe has the same diameter, \(\displaystyle A_1 = A_2 = A\) and since \(\displaystyle Q_3\), \(\displaystyle A\) and \(\displaystyle v_1\) are known,

\(\displaystyle v_2 = \frac{Q_3}{A} - v_1 \)

That's one less unknown, so you should be able to solve for \(\displaystyle p_2\) and \(\displaystyle p_3\).

You will get some head loss at the junction, but for an estimation you can assume that the head loss is negligible.

 

[pawel]

Sure, deriving velocities isn't a problem, but it's not enough to solve neither energy conservation for a whole system (unknowns: p2, p3) nor system of 2 energy conservation equations, each for particular stream (unknowns: p2, p3, deltaE - energy exchanged between streams). At least without an assumption that deltaE=0.

My point is that I don't understand why is it correct to assume so and use system of 2 Bernoulli's equations (first one for sections 1 and 2, second for 2 and 3), without any additional term. Is energy between 2 and 3 (or similary between 1 and 3) really conserved, despite influence of stream 1 (that, I think, should be treated as external source of energy, while considering stream 2 and it's "continuation" in stream 3, what Bernoulli's equation does)?

 

[pawel]

Sure, deriving velocities isn't a problem, but it's not enough to solve neither energy conservation for a whole system (unknowns: p2, p3) nor system of 2 energy conservation equations, each for particular stream (unknowns: p2, p3, deltaE - energy exchanged between streams). At least without an assumption that deltaE=0.

My point is that I don't understand why is it correct to assume so and use system of 2 Bernoulli's equations (first one for sections 1 and 2, second for 2 and 3), without any additional term. Is energy between 2 and 3 (or similary between 1 and 3) really conserved, despite influence of stream 1 (that, I think, should be treated as external source of energy, while considering stream 2 and it's "continuation" in stream 3, what Bernoulli's equation does)?