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Calculate flow rate in a multi-cylinder positive displacement pump

Calculate flow rate in a multi-cylinder positive displacement pump

Calculate flow rate in a multi-cylinder positive displacement pump

(OP)
I have been looking at a few technical articles and the image shown for the flow rate of a positive displacement pump doesn't match what I would expect. The image appears two have two 'humps' in the flow rate where I get regular peaks. can someone tell me if my understanding is correct.

The flow rate from one cylinder is :

  ksin(wt)*(sin(wt)>=0)

where: k=some constant
       w=angular velocity
       t=time

And for two cylinders each one is shitfed by pi (180°)

flowrate = (ksin(wt)*(sin(wt)>=0))+(ksin(wt+pi)*(sin(wt+pi)>=0))

And for three cylinders each one is shifted by 2/3 pi (120°)

flowrate = (ksin(wt)*(sin(wt)>=0))+(ksin(wt+(2/3*pi))*(sin(wt+(2/3*pi))>=0))+(ksin(wt+(4/3*pi))*(sin(wt+(4/3*pi))>=0))

Is this theory correct or am I missing something glaringly obvious?

RE: Calculate flow rate in a multi-cylinder positive displacement pump

I glanced at figure 1 page 1 and it looks right to me.  Perhaps you need to take a close look at the legend of the graphs.  The first one shows a one and three cylinder pump going through one and a half revolution.  The lower scale is in radians.  Piston speed is dependent on crank angle.  Note that piston must stop at TDC and BDC and change direction with the highest piston velocity midway between them.

RE: Calculate flow rate in a multi-cylinder positive displacement pump

(OP)
My formula above allows for the fact that there is only flow in the first 180 degrees (pi radians) of motion and that the maximum flow rate is at the mid point of the motion (90 degress or pi/2 radians) but I can't get my formula to agree with the chart.

I presume that the chart is correct because it isn't the first time I've seen the same plot.

RE: Calculate flow rate in a multi-cylinder positive displacement pump

(OP)
I've cracked it.

What I forgot to consider was the interaction of the connecting rod.

If y=crank throw(r)/connecting rod centres(l) then

If P=y.sin(wt).cos(wt)
And Q=1-(y^2).((sin(wt))^2))

Velocity V(w,t)=w.r.(cos(wt)+(P/(Q^(1/2))))

If you consider flow only occuring when the velocity is positive then

Flowrate Q(w,t)=A.V(w,t).(V(w,t)>0)

Where A is the area of the piston face.

This gives different curves dependant upon the crank throw and the distance between the connecting rod centres for the same angular velocity.

If there a n cyclinders this gives the following:

Total Flowrate=flowrate(wt) + flowrate(wt+z).(z>=n) + flowrate(wt+2z).(z>=n)....

where z is the phase shift between each cylinder.


.........I think

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