Bernoulli's equation: pressure and fluid velocity

[ihatebluescrubs]

I have a stupid question: bernoulli's equation is P+ pgh+ 1/2pv^2=K. When we are comparing two regions of a pipe, one with large area (smaller velocity), and smaller area (large velocity).

When we plug in velocity into the bernoulli's eq, shouldn't the pipe with a smaller area have a higher pressure? So shouldn't a region with faster flowing fluid have a higher pressure?

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[Mission Medical]

I have a stupid question: bernoulli's equation is P+ pgh+ 1/2pv^2=K. When we are comparing two regions of a pipe, one with large area (smaller velocity), and smaller area (large velocity).

When we plug in velocity into the bernoulli's eq, shouldn't the pipe with a smaller area have a higher pressure? So shouldn't a region with faster flowing fluid have a higher pressure?

Bernoulli's Principle actually tells us that the opposite, counter-intuitively, is the truth. A region with a smaller area does have faster flowing fluid (higher velocity), but the higher the velocity of the flowing fluid, the lower the pressure. One way to make sense of this is in terms of conservation of energy. Think of pressure as energy density. If velocity is increasing, then kinetic energy is increasing. To "compensate", the energy density (pressure) decreases. This is kind of like how, if somebody jumped off a cliff, his kinetic energy would increase, but his potential energy would decrease, thus conserving mechanical energy.

 

[ihatebluescrubs]

Thanks for the analogy, I see what you mean.

The equation is still confusing to me because the way it is, higher velocity should mean a higher 1/2pv^2 just like how a lower depth (larger h in pgh) would give a higher pressure.

Is there something I'm missing from the eq?

 

[GomerPyle]

Bernoulli's Principle actually tells us that the opposite, counter-intuitively, is the truth. A region with a smaller area does have faster flowing fluid (higher velocity), but the higher the velocity of the flowing fluid, the lower the pressure. One way to make sense of this is in terms of conservation of energy. Think of pressure as energy density. If velocity is increasing, then kinetic energy is increasing. To "compensate", the energy density (pressure) decreases. This is kind of like how, if somebody jumped off a cliff, his kinetic energy would increase, but his potential energy would decrease, thus conserving mechanical energy.

Actually - doesn't higher velocity through a pipe mean less kinetic energy (a decrease in randomness of fluid particles)...thus from KE = 3/2RT, the temperature also decreases...? I remember reading this in examkrackers.

 

[Mission Medical]

Actually - doesn't higher velocity through a pipe mean less kinetic energy (a decrease in randomness of fluid particles)...thus from KE = 3/2RT, the temperature also decreases...? I remember reading this in examkrackers.

I'm not sure about that, but my understanding was always that if velocity is increasing, the kinetic energy increases according to: KE = (1/2)mv^2

From HyperPhysics (With regards to the Bernoulli Principle): "This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy."

 

[GomerPyle]

I'm not sure about that, but my understanding was always that if velocity is increasing, the kinetic energy increases according to: KE = (1/2)mv^2

From HyperPhysics (With regards to the Bernoulli Principle): "This lowering of pressure in a constriction of a flow path may seem counterintuitive, but seems less so when you consider pressure to be energy density. In the high velocity flow through the constriction, kinetic energy must increase at the expense of pressure energy."

I don't think this is right. I remember that an increase of velocity within a pipe results in a decrease in randomness of the fluid particles (a decrease in kinetic energy) which causes a decrease in temperature (due to KE = 3/2RT) and the decrease in temp causes a decrease in pressure since volume is constant (via pv=nrt). This was talked about in EK and I believe in a practice test (don't remember which one).

 

[Vicodeen]

Random motion kinetic energy = Pressure

Translational kinetic energy = Velocity

That should clear your guys' argument.

 

[jsmith1]

I have a stupid question: bernoulli's equation is P+ pgh+ 1/2pv^2=K. When we are comparing two regions of a pipe, one with large area (smaller velocity), and smaller area (large velocity).

When we plug in velocity into the bernoulli's eq, shouldn't the pipe with a smaller area have a higher pressure? So shouldn't a region with faster flowing fluid have a higher pressure?

Ps = Psmall pipe & Pb = P big pipe

Ps + 1/2pv^2 + pgh = pgh 1/2pv^2 + Pb (the pgh's cancel out)
Ps + 1/2pv^2 = 1/2pv^2 + Pb

Since the velocity is larger in the smaller pipe Ps (pressure in small pipe) must be smaller so that the left side of the equation equals the right.

edit: small mistake in equation