Bernoulli's equation does not always work?

[theBEAST]

Homework Statement

The Attempt at a Solution

Alright so first I found a relationship between z1 (depth of inflow) and z2 (depth of outflow) using mass conservation. I found that 5 * z1 = z2 and I know this is correct because the answer key has the same relationship.

Next I decided to use bernoulli's equation to find another equation to relate z1 and z2. So using the streamline on the free surface:

P1 + 0.5 * rho * V1^2 + rho * g * z1 = P2 + 0.5 * rho * V2^2 + rho * g * z2

Since the streamline is on the free surface, P1 = P2 = Patm, so they pressures will cancel out and I am left with:

0.5 * rho * V1^2 + rho * g * z1 = 0.5 * rho * V2^2 + rho * g * z2

Plugging the numbers in, cancelling the rhos and substituting z2 = 5 * z1:

0.5 * (5)^2 + (9.81) * z1 = 0.5 * (1)^2 + 9.81 * (5 * z1)

Solving this yields z1 = 0.3058m...


HOWEVER this is not the same answer as the one given in the solution manual... The solution manual uses conservation of momentum. Why is it that bernoulli's does not work in this case?

 

[voko]

Hydraulic jumps dissipate energy. Bernoulli's equation is about conservation of energy. The two are not compatible.

 

[theBEAST]

Hydraulic jumps dissipate energy. Bernoulli's equation is about conservation of energy. The two are not compatible.

Why and how does it dissipate energy?

 

[voko]

The laminar flow is turned into turbulent; the velocity of 1 m/s is the average velocity of the flow, not the real (local) velocity anywhere of the fluid, which is much greater.

See http://en.wikipedia.org/wiki/Hydraulic_jump

 

[Nickie]

I would respond by saying that Bernoulli's equation is a simplified version of the more general conservation of energy equation. While it may work in certain cases, it is not always applicable and should not be relied upon as the sole method for solving fluid mechanics problems. In this particular case, conservation of momentum is a more accurate approach and should be used instead. It is important to understand the limitations and assumptions of any equation or model in order to accurately apply them in real-world situations.