Question 26 Bernoulli equations

[dahmsom]

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why is region one the greatest? According to the passage y3 is greater than y1 and y2 . I thought according to the equation a greater height will lead to a greater pressure? Am I seeing this wrong ?

 

[K9TestRun]

you are referring to question 26, correct?

 

[dahmsom]

@ElApacheNess yes

 

[K9TestRun]

If velocity doesn't change, you get P1+pgy1=P2+pgy2, and since the quantity pgy2 is greater, P2 must be less than P1 for the equation to hold true.

 

[sazerac]

Except the problem states y1 = y2.

What is a constant is not the velocity, but rather the flow rate = volume / second = area x velocity. Otherwise, every second that goes by, material would need to be created (or destroyed).

The fluid must be moving faster in section 2 because it is a smaller pipe. Think about when you put your thumb over the end of a hose... smaller hole means faster water.

Bernoulli says faster things have less pressure. Because they spend more of their energy moving in a straight line they have less energy to be beating on the sides of the container (which is what pressure is).

 

[theonlytycrane]

Bernoulli's eqn describes the conservation of energy for fluids. The 3 components of energy to think about are:

Pressure = P1, P2, P3 (depending on the region)
Potential energy = Density * g (gravity) * y (height)
Kinetic Energy = 1/2 * Density * v^2

Fluid flows through the pipe, and though the cross-sectional area of the pipe or height of the pipe changes in different regions, the overall energy of the fluid remains constant. Also remember to use the continuity equation a1* v1 = a2 * v2 to compare how velocity changes as cross sectional area changes between regions.

For example, when comparing region 1 to region 2: Let P1 = Pressure of region 1, D1 = Density, v1 = velocity, A1 = area, etc.

P1 + D1 * g * y1 + 1/2 * D1 * v1^2 = P2 + D2 * g * y2 + 1/2 * D2 * v2^2

y1 = y2, so the Potential energy is the same in these regions.
A1 > A2, so v2 > v1 for the continuity equation to hold.
Using bernoulli's eqn above, since v2 > v1, P1 > P2 for energy to be conserved. region 1 and region 2 have the same potential energy. Region 2 has higher kinetic energy, so region 1 must have higher Pressure.

Now comparing region 1 to region 3 follows the same logic, except now A1 = A3 so v1 = v3 (continuity equation).
y3 > y1 (region 3 has more potential energy), so P1 > P3.

Bernoulli's eqn and the continuity eqn are all you need to compare fluid flow in different regions. Hope this helps!