Calculating pressures in a system with a manifold and changes in areas

[USAeng]

If air is pumped in with a fan at a flow rate Q then split 3 ways (Q1,Q2,Q3) all with identical cross sectional areas - then the velocities and pressures will be the same across the board. I get that, but I cant find equations that explain what happens after one of the areas change...

I can apply the continuity equation at each section to see how the velocity and cross section area relate to the flow rate...

I assume that by reducing the cross sectional area at section 3 by 50% then the flow would change at sections 1 and 2 as well...

Does anyone have a pdf of a good practical example in a book that they could give me to explain this situation? I know it has to be pretty simple... I just cant remember how this worked from back in college... Thanks a lot

The application is actually going to be using a pressure switch or differential pressure switch to determine when one or more of the sections have become blocked by more than a certain percentage.

I have attached a little picture of the simple problem.
Thanks a lot for any help.

 

[zdas04]

[I don't open .png files, they cause me to have to re-install Acrobat. I'm going to assume you are going from a header to a header or to atmosphere, all pipe lengths are the same and they dump into a common sink]

To begin, Q1 will never equal Q2 or Q3, none of the flow rates will be the same. Ever. You have entry effects, varying surface roughness, and exit effects. These can be big, or they can be small, but they are never zero. The flow rates can be close, relying on them being exactly equal is a mistake.

Next, it is anything but simple. A flowing gas can only go from a high pressure to a lower pressure. A microbar of difference with change flow distribution. A screw protruding into the conduit sets up von Karmen streets that dramatically change flow distribution.

For equal size (and length) conduits, we pretend that they'll be the same because there is no way to predict the actual flow distribution. For conduits of differing sizes and/or lengths, most of us solve the problem iteratively. Take the header pressure, exhaust pressure, pipe size and length and use an empirical equation like the AGA Fully Turbulent equation (or you can use Weymouth or Panhandle A) to calculate the flow in each pipe. The sum of the three pipes will not equal the total available flow, so adjust the pipe roughness (in AGA) to try to account for entry/exit effects. Iterate until your three pipes use up the mass flow rate in the header. That gives you a rough estimate of the distribution, but when you turn it on the actual will be different.

David Simpson, PE
MuleShoe Engineering

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[BigInch]

There are no equations to proportion flow amongst branch outlets based on cross-sectional area, because flow into each branch is not controlled by the cross-sectional areas alone. Flow is proportional to differential pressure in all downstream flow elements, or resistance if you will, and resistance is (basically) proportional to both flowrate divided by cross-sectional area, as well as a few other things.

A joint is theoretically only a point in a big piping system, thus theoretically has no length and therefore no differential pressure itself. Differential pressure at a joint is provided by the driving pressure in the upstream piping and by the back pressure in the downstream piping. Flow into each branch would only be proportional to only the cross-sectional areas of each branch when the outlet pressures just downstream of the joint in each branch were maintained exactly equal, something which is not happening very often.

What you are considering, restricting flow in pipe 3, would redistribute flows into branches 1 and 2 by increasing the resistance to flow in branch 3, thereby increasing the pressure drop of branch 3 in relation to 1 and 2. Flow always seeks the path of least resistance, thus more and more flow is diverted away from branch 3 and forced into branches 1 and 2 until the resistance (flow/area) of all downstream branches are once again equal.

So, to calculate the flow distribution at the branch, the flow resistance (head loss) of each downstream branch must be equal. You therefore need to write a head loss equation (there are a few you could use, Darcy-Weisbach, Manning's, Colebrook-White, I suggest Churchill) for each branch and solve those 3 equations simultaneously, or by iteration, for the flowrate in each joint.

Or obtain some software, EPAnet (is free), to do it for you.


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[sheiko]

zdas04 and BigInch are once again correct. Flow will distribute to equalize pressure drop in each path.
A methodology for solving flow-splitting problems is proposed in the following link:

http://kb.eng-software.com/questions/378/Value+of+System+Resistance+Curve+(Part+3)

"We don't believe things because they are true, things are true because we believe them."

 

[USAeng]

Ok... so it would definately simplify things to measure the pressures and velocities on site rather than calculate...

We are concerned about one or more of the 3 sections getting slagged over more than a certain percent. My original plan was to install pressure switches... but then I realized that if one section got closed off at the end- that the static pressure would equalize... also, if all 3 got slagged over a small amount the static pressure would be the same as 1 being mostly slagged over and the other 2 open.

So my new thought is to use velocity pressure and measure with a pitot tube and have that hooked up to a pressure switch. We will have 1 tube and 1 switch on each section (6" ducts) and when the end of one gets closed off a certain amount the velocity back where the pitot is will go down and trip the switch...

Sorry if I made it seem like I wanted to calculate everything... I guess my real question should have been more focused on the application of the pressure switch

Thank you for all your thoughts so far... so far I am looking at Dwyer Instruments for everything... if anyone has any thoughts or suggestions I really appreciate it.

Have a good one.

 

[1gibson]

Do you want to calculate it, or measure it?

How about mass flow sensors in each branch (possibly too restrictive or expensive.) Or this contraption that just popped into my head:

A rod that is hinged at the pipe OD to allow movement in the direction parallel to the pipe/flow. Lightweight rubber seal or something of that sort that will prevent leakage. There is a flat element on the end of the rod, protruding into the flow, and a scale or marker on the other end of the rod outside of the pipe.

Flow changes the angle of the element from straight down to tilted in the direction of the flow. Come up with appropriate range of movement based on weight, surface area, counteracting spring tension, etc.

3 pipes with identical cross sections (and enough straight pipe for flow to re-establish after the reduction), 3 identical elements, 3 scales with arbitrary graduations on the marker from 0 to "full flow." If one is not like the others, there is a flow restriction.

$100 and a trip to the hardware store. Potentiometers attached to marker if you need remote monitoring, +$50.

?