Effusion Rate = sqrt(2gh). But what about Area?

[justadream]

“Students must create an irrigation system that takes water from a reservoir 80 cm deep to a wave pool across the room. A perfectly leveled, horizontal tube with constant circumference takes water from the bottom of the reservoir to the wave pool.

If the tubing used has a cross-sectional area of .02m^3, what is the velocity of the water as it enters the wave pool?”

The answer is found using V = sqrt (2 *g * height). I understand how to get this equation.

My question is: Intuitively, why doesn’t the cross-sectional area of the opening matter? For example, if the cross sectional opening were 5 cm wide as opposed to 1 cm wide? The continuity equation (A1V1 = A2V2) would imply that the cross-sectional area does affect velocity.

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[Cawolf]

Toricelli's theorem is a derivation of Bernoulli's equation assuming equal pressures and a velocity in the large pool that is approximated by 0. This velocity is not dependent on area as the terms in the equation account for energy from pressure, kinetic, and potential. The velocity does not depend on area as shown by this proof. It is derived from the kinetic energy of the fluid.

 

[justadream]

@Cawolf

A later question on this asked about the velocity of the water as it exits the tube (not as it enters). As it exits the pool, the radius is different from when it entered. Thus, cross-sectional area suddenly matters now.

That means:
1) When it enters the tube from the big tank, cross-sectional area doesn't matter
2) When it exits the tube (after the tube has changed radius), cross-sectional area does matter

So when exactly does cross-secitonal area start "mattering" for velocity then?

 

[Cawolf]

Like the solution I posted in the other thread just a moment ago. I used the ratio of area and velocity to develop a term to plug into bernoulli's equation.

When using Toricelli's theorem you are approximating the velocity of the water in the tank to be zero.

Area matters for example in a hose. If you put your thumb over the tip and make the hole smaller, the water leaves the hose at a higher velocity.

 

[Chrisz]

“Students must create an irrigation system that takes water from a reservoir 80 cm deep to a wave pool across the room. A perfectly leveled, horizontal tube with constant circumference takes water from the bottom of the reservoir to the wave pool.

If the tubing used has a cross-sectional area of .02m^3, what is the velocity of the water as it enters the wave pool?”

The answer is found using V = sqrt (2 *g * height). I understand how to get this equation.

My question is: Intuitively, why doesn’t the cross-sectional area of the opening matter? For example, if the cross sectional opening were 5 cm wide as opposed to 1 cm wide? The continuity equation (A1V1 = A2V2) would imply that the cross-sectional area does affect velocity.

K=P +pgh+0.5pv^2 big P pressure, small p density. During the derivation of Bernoulli, cross section area is taken into account, but eventually arrive the equation in terms of density. I am typing with ipad, so I won't waste time write out the derivation. But for a continuous pipe or flow path way of any shape, Bernoulli holds.

P1+pgh1+0.5pv1^2=P2+pgh2+0.5pv^2
1 represent at top of tank, 2 represent at time when water exists

P1=P2= ATM pressure, because at top of tank, it has a extremely large area compared to openning, so v1 is small, so pg(delta h) =0.5pv^2, little algebra leads to the answer

 

[DrknoSDN]

@Cawolf
So when exactly does cross-secitonal area start "mattering" for velocity then?

Area does not matter when gravity is the driving force. If gravity is producing a pressure that is a force/area then it doesn't matter because as you increase the area you will just have more force to push more fluid.

This is contrary to in a horizontal pipe. In a pipe the only reason the fluid continues to flow is because the fluid behind it (earlier in the system) is pushing into it. For this reason if you had a large water tank with a 1 cm^2 area horizontal pipe connected you would use the V=root(2gh) equation to determine velocity at the tank/pipe connection. If the pipe later expanded to a 20cm^2 area the change in area would cause a decrease in velocity because the limiting factor is how much flow is occurring due to gravity at the tank/pipe connection.