If a fluid is passing through a one inch pipe which gets necked down to 3/4" and then back up to 1", the pressure and volume will be the same at the output, correct? I am trying to get a good volume of air to my forge and am experimenting with some tweaks to my inadequate setup.
Thanks,
Rob
The same as what? The same as upstream of the neck, or the same as it was before you installed the neck?
If the former, the answer is no, although the difference may be negligibly small, depending on the flow rate. If the latter, the answer is “yes,” but the pressure upstream will be higher than it was previously (and flow rate depends on the flow source).
From the way I’m reading the OP, it seems the pressure at the outlet of the pipe will be equal to the inlet pressure minus the pressure drop from the flow effects at the transition points, as well as the frictional losses (assuming this is a steady-flow system and not a static system). It’s been 18 or more years since I took Fluids in school but I do remember running through calculations on complex water and air pipes and accounting for all the pressure drops along the way.
Here is a nice little equation for Venturi meters (which is similar to your setup).
If you’re comparing input to the pipe with output from the pipe, then:
The flux (volume of air passing through) pretty much has to be the same at the input and output of any pipe. The pressure will always be at least a tiny bit lower at the output (otherwise, the air wouldn’t be flowing).
If you’re comparing output-without-a-neck versus output-with-a-neck, then the answer is that the output with a neck will be at least a tiny bit less (both pressure and flux), but whether it’s a significant amount less is going to depend on the specifics of the situation.
An engineer could probably tell you, if you specified the pressure/flow at the input to the pipe and some basics on the geometry. IANA flow engineer, but I’d guess that you could tell the difference: a 3/4" diameter pipe is only about half the cross-section of the 1" pipe. That’s probably going to slow things down.
Best bet is really to experiment both ways, if possible.
Two engineers have already told him. 
(that is intended to be light-hearted not snarky)
I think venturi meters detect the difference between the air approaching the venturi and the air in the throat itself. Obviously the pressure in the throat must be lower than in the approach, because otherwise air wouldn’t accelerate as it approached the throat so its velocity could be highest in the throat.
But it is often important in the design of venturis to recover that pressure as the air exits. It must be tapered gradually so turbulence does not develop. The air will slow down as it leaves, because it is climbing a pressure gradient, and most of the pressure will be recovered if it is designed nicely.
You can figure that it takes a pressure drop of 1 inch water column to get air going 4000 feet per minute, and the pressure changes as the square of the velocity (gee I hope I remembered the numbers right).
You can also experiment with water if you want to be able to see what is going on. The kinematic viscosities of air and water are pretty similar at room temperature, so they will swirl the same way going through the same geometry at the same speed. The pressures, though, will be very different, so this is mostly useful as a visualization method.