Q & A: bottlenecks to flow

I'm a welder and mechanic who's working on building a shop building with a couple of other welder/mechanics. We've hit the same question twice with electricity and air lines: Is flow going to be determined by the smallest fitting or wire in the system? One of us (me)says yes. One says that one small fitting or one small length of smaller wire won't have as big an effect as if we had dropped size on the whole system or a long length. One guy just says it doesn't matter except for heat breakdown danger with the electricity because we have the same push (voltage in one case and a 160 psi compressor in the other). Who's right?
- Eric (age 40)
Pueblo, CO USA

You're all partly right. For a given pressure or voltage the flow is determined by the total resistance. The resistance comes from adding up the resistances of all the parts of the line in series. Narrow necks can be the dominant contributor, but they aren't the only contributor. Especially for laminar flow of fluids, where the resistance goes as the inverse of the fourth power of the pipe radius, even a short narrow segment can make a big difference. For electrical current, the resistance only goes as the inverse of the square of the wire radius.

Definitely there's a big problem with even a small narrow segment in an electrical circuit, due to the heating, proportional to I²R, where I is the current running around the circuit. If there a small segment with high R, lots of heat gets dumped in that small region, which creates obvious serious safety issues.

I guess some of the beauty of having a mathematical way to describe these things is that it lets you assign importance to the various parts without switching into all-or-nothing verbal approximations.

Mike W.

(published on 05/03/2011)