Air flow through increasing pipe size

Air flow in shop is through 1/4" hose. pneumatic motor has 1/2" inlet plus the tubeing to and from valve is also all 1/2" tube. what would be a general equation to model the pressure loss or increase to the motor.

Shop air:
80psi 60cfm

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So looking back a bit or lot more paper education would have been good for me but, I say the pressure drops for sure (put thumb over garden hose it increases). But, how much, don't know. I would guess double the pipe you half the pressure.

I am pretty flighty with my passions. So I got it in my head to use my natural math (just above zero to many of you I'm sure) ability, the generic teaching system and this here internet to grow my understanding to at least where I can understand what is being said in those beautiful numbers. I might be a bit of a pain on here for a bit.

"I would guess double the pipe you half the pressure. "

But don't quote me on that...

well i know when you decrease the pipe size and volumetric flow rate remains constant - velocity increases and pressure drops, i would assume the inverse is true with a distinct loss of energy but i'm unsure as to the driving equation as every "online calculator" throws a fit if you go small to big versus big to small.

when you decrease the pipe size and volumetric flow rate remains constant - velocity increases and pressure drops,
Really? Same volume going through smaller pipe. I can't get my head around this quite yet.

take it like 4 lanes of cars going down to 2 lanes and the cars are invencible and the cars continue to come at a steady rate, basically for the system to work the cars moving into the 2 lane section will have to move faster. that analogy handle the mass but not the pressure but you get the idea right?

If you want a governing equation, Bernoulli's law covers this for non-extreme flow rates.
$$P + \frac{1}{2}\rho v^2 + \rho g h = constant$$
In your case the only variables changing from one tube to the other are the flow velocity and the pressure, both of which you can work out from your pressure and tube diameter data if you also know the volumetric flow rate..

Q_Goest
Homework Helper
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what would be a general equation to model the pressure loss or increase to the motor.
Pressure drop through pipe or tube is generally done per the Darcey Weisbach equation which is given by equation 1 in the attachment called "Pipe-Flo Pro.PDF" https://www.physicsforums.com/showthread.php?t=179830".

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Thank you for the info on the document i think it may be invaluable to me in the futrue, however right now i don't believe the DW equation is what im looking for, unless there is a way of restating it that works for my case.

I'm trying to get the dp with the sudden change of diameter. Length is not my conern right now.

I have read throught the entire document yet so the answer may yet be in there but thank you any way because i know i'll use it in the future.