What causes the decrease in pressure in a smaller diameter tube?

[nautikal]

So we are learning about fluid dynamics in my physics class right now, and while I "know" the concepts, I don't understand them.

Bernoulli's law says that when a fluid flows from a large area to a smaller area, the pressure decreases and the speed of the fluid increases. I understand how the speed increases - the flow rate must remain constant (assuming an incompressible fluid). But the pressure decreasing seems counter intuitive to me. The ideal gas law states that decreasing the volume of a container increases the pressure. When a fluid moves from a larger diameter tube to a smaller one, why then does the pressure not increase as the ideal gas law would predict?

I'm also confused about flow rates and energy. Say we have a vertical pipe with a constant diameter with water being pumped through it. The flow rate at the top and bottom are equal, and therefore the velocity of the water at the top is equal to that at the bottom. Yet this means the potential energy increases while the kinetic energy remains constant. How is this possible if energy is conserved? Bernoulli's law says that the pressure at the top will be less than at the bottom, so is pressure some sort of energy (indeed its units dictate that it's a measure of energy per volume)?

 

[rcgldr]

Bernoulli's law says that when a fluid flows from a large area to a smaller area, the pressure decreases and the speed of the fluid increases. I understand how the speed increases - the flow rate must remain constant (assuming an incompressible fluid). But the pressure decreasing seems counter intuitive to me.

This is an idealized case, the fluid and pipe are frictionless, there's no viscocity (friction within the fluid), and the fluid is incompresseable. The pipe peforms no work on the fluid (something is performing work on the fluid in order to maintain a flow at a specific pressure at one end of the pipe, but this doesn't matter for what happens inside the pipe where no work is done).

Since no work is peformed by interaction between pipe and fluid, then the only cause remaining for the acceleration or deceleration of the fluid is a pressure differential corresponding to the rate of change of pipe diameter. The faster moving fluid in the narrower section of pipe must have lower pressure in order for the higher pressure of the slower moving fluid in the wider section of the pipe to accelerate.

In real life, friction, and viscosity cause the pipe to perform work on the fluid, and a flow in a long pipe with a constant diameter will decrease as it flows towards the exit end of the pipe. In spite of this, a tapered conic like section of pipe will result in increase in speed and reduction of pressure, usually called Venturi effect.

Say we have a vertical pipe with a constant diameter with water being pumped through it. The flow rate at the top and bottom are equal ... kinetic energy

Bernoulli principle states that the total energy of a fluid is the sum of it's pressure, kinetic, and gravitational potential energy. The pressure of the fluid would increase as it descends in a pipe (this requires that the pipe exit pressure equals the fluid pressure at that point in the pipe). The only energy being ignored here is temperature.

 

[russ_watters]

The ideal gas law states that decreasing the volume of a container increases the pressure. When a fluid moves from a larger diameter tube to a smaller one, why then does the pressure not increase as the ideal gas law would predict?

A decrease in cross sectional area is not a decrease in volume. Area and volume are two different things.

I'm also confused about flow rates and energy. Say we have a vertical pipe with a constant diameter with water being pumped through it. The flow rate at the top and bottom are equal, and therefore the velocity of the water at the top is equal to that at the bottom. Yet this means the potential energy increases while the kinetic energy remains constant. How is this possible if energy is conserved? Bernoulli's law says that the pressure at the top will be less than at the bottom, so is pressure some sort of energy (indeed its units dictate that it's a measure of energy per volume)?

Yes, Bernoulli's principle is a manifestation of conservation of energy.

 

[nautikal]

Bernoulli principle states that the total energy of a fluid is the sum of it's pressure, kinetic, and gravitational potential energy. The pressure of the fluid would increase as it descends in a pipe (this requires that the pipe exit pressure equals the fluid pressure at that point in the pipe). The only energy being ignored here is temperature.

Okay, but how is this explained on the molecular level? An individual molecule of water will have the same kinetic energy as it goes up the pipe at a constant speed, yet its potential energy will increase. There is no pressure at the molecular level, so how is this explained? Is pressure a type of potential energy at the molecular level (since molecules are pushed together)?

A decrease in cross sectional area is not a decrease in volume. Area and volume are two different things.

Ahh... that makes sense. So if we have a gas moving from a larger diameter pipe to a smaller one, the gas molecules will actually be more spread out in the smaller diameter pipe (and thus the lower pressure)? Is it correct I think of this in terms of the following analogy? When there's a traffic jam cars move slowly and are closely bunched together until they pass the source of the traffic jam (ie an accident) where they then speed up and the cars move farther apart.

My chemistry is pretty strong, so it helps me to think on the molecular level.

 

[Nirgal]

I am very limited in my knowledge but I hope that I can help.

I do not believe that water can be modeled as an ideal gas because it is an incompressible fluid and ideal gasses are systems of particles that do not interact with each other.

The pressure is a measure of the weight of the fluid spread out over an area. Imagine two identical swimming pools. One is filled half way and the other is full to the brim. If you were to swim to the bottom of each pool you would find that there is more pressure on your ears in the pool that is full. This is because there is more water (or weight) pushing down on your head. If you were to shrink the diameter of the pool that is half full so that the water inside of it is raised to the same level of the pool that is full, then you would find that both pools would have the same pressure at the bottom. This is because, over a unit area, they would both have the same amount of water or weight.

The ideal gas law views the pressure through a different lens. It does not see pressure as the accumulated weight of a mass of particles pushing down over an area. It views pressure as a transfer of momentum from the particle to it's environment or container. Pressure is the amount of force over an area, so each particle of the ideal gas slams into the side of a container carrying a force, and that force is transferred into the container over a tiny area. The container is rigid so this force is transferred back to the particle like a perfect elastic collision, like a bunch of perfect rubber balls that do not loose any energy to friction as they bounce.

A fluid that flows through a large diameter of a pipe has a large pressure because there is a large mass contributing to it's weight. Just as gravity pulls on the mass of the water in the swimming pool, stimulating a pressure, the velocity of the moving fluid pushes it's mass along, stimulating a pressure that spans horizontally.

Within the smaller pipe there is less pressure because in a given volume at a frozen instant in time there is less mass. Over a span of time there is equal mass because the velocity of the fluid in the smaller pipe is greater. This is why mass is conserved.

Therefore for an incompressible fluid, I do not see pressure as a measure of energy, though you can convert its units into such. It is an amount of mass pushing against an area with some speed. Its like a continuous transfer of force.

 

[rcgldr]

Bernoulli principle states that the total energy of a fluid is the sum of it's pressure, kinetic, and gravitational potential energy. The pressure of the fluid would increase as it descends in a pipe (this requires that the pipe exit pressure equals the fluid pressure at that point in the pipe). The only energy being ignored here is temperature.

Okay, but how is this explained on the molecular level?

I'm not sure it can be when using an abstract incompressable, inviscid, fluid for this thought experiment. An incompressable, inviscid, fluid would be made of magic polygon shaped incompressable molecules that have no gaps between, while at the same time the inviscid (zero viscosity) aspect would allow free movement of these magic molecules with respect to each other without introducing any gaps. The speed of sound in an incompressable fluid would be infinite. The pipe or any container used for this thought experiement would also have to be incompressable and frictionless.

If you repeat the experiemet with a real fluid, then things change, mostly some amount of resistance to flow by the pipe due to friction, viscosity, and turbulence. In a no flow situation, the pressure of water increases with depth, along with a small increase in density versus pressure. The downforce on gravity on all the molecules above a certain level ends up as downforce on the molecules at that certain level, which causes the increase in pressure. In the case of the atmosphere, at around 1940 feet of altitude, the pressure decreases from 14.7 psi to 13.7 psi. In a 1 square inch column of air, there's 1 pound of compressed air from zero to 1940 feet altitude, and 13.7 pounds of air from 1940 feet to the outer edge of the atmosphere.

 

[russ_watters]

Okay, but how is this explained on the molecular level? An individual molecule of water will have the same kinetic energy as it goes up the pipe at a constant speed, yet its potential energy will increase. There is no pressure at the molecular level, so how is this explained? Is pressure a type of potential energy at the molecular level (since molecules are pushed together)?

Sounds like you're figuring it out on your own just fine. Yes, pressure is essentially a type of potential energy with some simplifications and Bernoulli's equation has a term for static pressure (in the first rewritten form): http://en.wikipedia.org/wiki/Bernoulli's_principle#Incompressible_flow_equation

Ahh... that makes sense. So if we have a gas moving from a larger diameter pipe to a smaller one, the gas molecules will actually be more spread out in the smaller diameter pipe (and thus the lower pressure)?

Sorta - they'll spread out along the streamline, which explains the increase in speed, but the density doesn't change. Bernoulli's principle (the basic version, anyway) is for the assumption of constant density and therefore small changes in pressure.

Is it correct I think of this in terms of the following analogy? When there's a traffic jam cars move slowly and are closely bunched together until they pass the source of the traffic jam (ie an accident) where they then speed up and the cars move farther apart.

Not bad, it's just that people don't really act the same way that Bernoulli's implied - if they did, it would solve traffic problems because they'd speed up at choke points to avoid an overall reduction of traffic flow. Instead, it's like friction and a loss of energy...

 

[rcgldr]

How is this explained on the molecular level? Is pressure a type of potential energy at the molecular level (since molecules are pushed together)?

The molecules have an average kinetic energy (1/2 m v^2) related to their Kelvin temperature. The pressure is the result of molecular collisions with the (impulse of momentum change / unit time), applying an average force per unit area to the walls of a container, or with each other, and this is called static pressure.

If the total energy of the molecules isn't changed (no work done), but their speed in a particular direction is, then their movement is a bit less random, with a net increase in movement in the direction of flow, and a net decrease in movment perpendicular to the flow. This requires some form of acceleration that doesn't involve work, such as a ideal frictionless pipe with smoothly decreasing cross sectional area to avoid turbulence.

Since there is a net flow due to acceleration with no change in total energy, the randomness of the velocities of the molecules is decreased, and so is the average difference between velocities of the molecules, decreasing the collision related impulses / time, which decreases the force per unit area, or static pressure.

If some measuring device decelerates a small portion of the flow back to it's original state, then it can measure the original total energy, all as static pressure within the device. The pressure within the device minus the static pressure of the flow is called dynamic pressure. Total pressure = dynamic pressure + static pressure, and if no work is done, the total pressure is constant.

An individual molecule of water ... as it goes down the pipe at a constant speed ... its potential energy will decrease.

I changed this for water flowing down a pipe at constant speed. Assuming no work done other than by gravity, the kinetic energy of the molecule will increase as it's potential energy decreases. Since the flow speed is somehow kept constant, then the increase in kinetic energy is random instead of organized, the collision related impulses / time increases, increasing the static pressure. In order for the flow to be constant, the pressure has to increase by the same amount that potential energy decreases (or vice versa if an upwards flow). Again this is assuming an idealized frictionless pipe.

An idealized example of this would be a frictionless oval shaped pipe, oriented vertically (horizontal axis), with water circling through the pipe only due to it's momentum. The pressure would vary the same as if the water wasn't moving (except for the centripetal related forces from the pipe on the curved sections).

So if we have a gas moving from a larger diameter pipe to a smaller one, the gas molecules will actually be more spread out in the smaller diameter pipe (and thus the lower pressure)?

For the idealized incompressable fluid case, the density is independent of pressure, so the density remains the same even though the pressure decreases. Relaxing the conditions to allow for a compressable fluid, but assuming a frictionless, incompressable pipe, the density decreases as the pressure decreases, and the flow in the smaller section of the pipe will be faster and the pressure even less (in order to have accelerated the fluid to a higher speed) in order to maintain the mass flow at a lower density. The only constant in both cases is mass flow per unit time through any cross section of the pipe.

The "Bernoulli" math for compressable fluid is a more complex than for the incompressable fluid, which is why incompressable fluid is often used for thought experiments. It's covered here in this wiki section:

http://en.wikipedia.org/wiki/Bernoulli's_principle#Compressible_flow_equation

Again, all of this interaction is done in a work free environment. There is no change in total energy. In the real world, virtually every conversion of energy from one form to another involves some amount of work. Pipes aren't frictionless and oppose any flow. In a real world pipe with constant flow, the static pressure gradually decreases over distance in the direction of flow, because the pipe peforms work opposing the flow.