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I was thinking of doing something with a flow sensor and a solenoid valve when I realised I was perhaps over-thinking it: is there perhaps such a thing as a purely mechanical 4-port valve consisting of two isolated lines, where water flowing on the A/B line opens a valve allowing water to flow on the C/D line, somewhat analogously to an electric relay?

A ]===°===[ B
      ^
C ]===%===[ D

It seems like it should be possible with ball bearings and a springs(?) so I'm just wondering if it exists, and if so what it's called so that I can see if I can find a (not outrageously priced) suitable one for the project.

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    Something like fluidics, but without violating the Ghostbusters rule about crossing the streams? The problem lies in detecting and using flow rather than pressure on the control side of the device. You can convert flow to pressure by adding a restriction, but that may impact the flow to an unacceptable degree.
    – HABO
    Commented Jul 14 at 13:52
  • @HABO well in logic terms I want simply A, i.e. the controlled side is on iff the controlling side is, but I think I see your point about pressure vs flow. The controlling side would be between tap (aka faucet) and its supply, so I suppose pressure drops slightly when the tap opens?
    – OJFord
    Commented Jul 14 at 14:21
  • Is it possible? Sure; two water wheels on a single shaft would sorta do it, matching the delivered volume. Or you could use a thermostatic mixing valve rather than relying on either pressure or volume. Not sure I understand what actual problem you are trying to solve so I'll leave it at that.
    – keshlam
    Commented Jul 14 at 16:35
  • @keshlam Yes that would do it as the internal mechanism I think. What I'm trying to do is buy something off the shelf that I can use to turn 'on' one tap when another is manually opened, essentially. (Not literally that, but a lot of irrelevant detail that boils down to that.) Start running warm water in the sink, cold water also flows elsewhere.
    – OJFord
    Commented Jul 14 at 21:06

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Yes, it's called a Pilot Operated Valve, featured in the Merrimack Valley gas explosions, Three Mile Island, and it dramatization in The China Syndrome movie, released 12 days before the event!

All those pilot-operated valves were being used as a feedback mechanism largely using the line they controlled as an input. However, the input could be independent.

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