Last time, we examined a puzzlement. Suppose a contractor attaches a 4” pipe to the water main and has a 4” pipe going into an apartment building, but connects the two larger pipes with a length of smaller 2” pipe in between to save money where it doesn’t show. Now suppose there is some flow through the pipes, but not enough to tell the difference between the pressures at both ends. The pressure at each end is 60 lbs/sq.in. The question is what will be the pressure in the small pipe.
Most will guess that the pressure in the small pipe will be larger. After all, they may reason, the sides of the pipe must squeeze the water into a smaller area. That answer imagines that the sides of the pipe do work on the water.
The answer, however, was discovered by the Swiss mathematician Daniel Bernoulli (1700 – 1782) some 250 years ago. He found that, if no external work is done on a moving fluid, an increase in the velocity must be accompanied by a decrease in the pressure within the fluid.
A good way to see why this must be so was suggested by my friend Lew Epstein in his Thinking Physics (ISBN 0-935218-06-8) in 1979. Imagine that a toy submarine is drifting with the water in the pipe. Think in terms of what the submarine must do to stay up with the water as it passes from section to section in the pipe.
(Lew’s submarines have little weights inside suspended by springs, so they are little accelerometers. I’m leaving out the weights for simplicity.)
If the flow rate, in gallons per minute, is to remain constant in the pipe, then the water in the smaller pipe must surely flow with a greater velocity. In fact, you can figure out exactly how much faster by considering that area times velocity must be constant. (This is done in more detail in class, where I have a blackboard.)
To go faster, the water and the submarine need more pressure behind it than in front of it. To slow back down, the water and submarine need more pressure in front of it than behind it.
To figure out how much the pressure must go down or up when the velocity goes up or down, we need to consider that no external work is done on the fluid so that the energy per unit volume remains constant. That assumes no fiction or turbulence – a severe limitation in the real world but it makes the underlining principles easier to think about.
Now if you think about pressure in terms of work and energy, you find that pressure in a fluid is really the same thing as energy density. Pressure we defined as force per unit area. Multiply numerator and denominator by distance perpendicular to the area, and we have work over volume.
P = F/area = F.d/ area.d = Work/Volume
Remember that we found the fluid pressure rule by expressing the weight of the fluid in terms of its weight density.
P = F/area = Wt/area = (weight density).(Volume)/(Area) = ρgh
We can think of the first equation as expressing the elastic potential energy density of the fluid and the second as the gravitational potential energy density of the fluid. We have done static problems calculating the water “head” needed to produce water pressure at the faucet when the fluid is not moving.
Now we can generalize this reasoning for a moving fluid by thinking of the mass and energy of the submarine. Let’s say the submarine has a mass m, a velocity v1, and a volume V as it is moving along in the first pipe. The difference in fluid pressure creates a force F that gives the submarine an acceleration a over a distance d. But now we have an added tool of kinematics that states that v22 – v12 = 2 a.d. We say that the kinetic energy Ek = (1/2) mv2 has changed with the velocity and the pressure or energy density has likewise changed.
Again, I can do this better on the blackboard, but an outline of the derivation is
That is sometimes called the Bernoulli term. It accounts for the kinetic energy of the submarine or, which is the same thing, a chunk of water -- and, since we have divided by the volume, it doesn't matter how small.
The full Bernoulli equation is sometimes written
P1 + ρgh1 + (˝)ρv12 = P2 + ρgh2 + (˝)ρv22
This shows an energy balance in each part of the pipe. It assumes the total energy density, including elastic potential, gravitational potential, and kinetic energies, are the same in each part of the pipe.
It takes care of changes of height as well as changes in velocity. It does not account for friction or turbulence, so it must be used with care.
