I fought the law
And the law won

("I Fought the Law," written by Sonny Curtis in 1958, recorded by The Clash in 1979)

 

THE THREE RECIPROCATING-ACTION PD PUMP "LAWS"

Centrifugal Pumps have the "Affinity Laws" (featured in PTOA Segment #169) so why wouldn't Reciprocating-Action PD Pumps have their own set of "Laws" related to Capacity/Flowrate?

The three Reciprocating-Action PD Pump "Laws" are all common sense statements that express how the internal hardware of the Pump impacts the Capacity/Flowrate discharged from the pump.

Brilliant PTOA Readers and Students  … meaning those who are reading the PTOA Segments in the intended, sequential order …  just learned in PTOA Segment #209 that every pump has a Pump Name Plate that lists the Pump's Capacity/Flowrate.

So like, who needs to know the Reciprocating-Action PD Pump Laws?

Why not just look at the Pump Name Plate to get an idea of the Pump's Capacity/Flowrate?

Excellent questions!

For the purpose of this PTOA Segment #210 pretend that there is no Pump Name Plate affixed to the Reciprocating-Action PD Pump.

In that case:

The first two Reciprocating-Action PD Pump "Laws" are:

The Capacity/Flowrate is proportional to the speed of the Driver.

and

The Capacity/Flowrate is proportional to the number of "strokes" of the Piston, Plunger, or Diagram per unit time.

Who noticed that those two "laws" are just common sense?

The greater the Driver's rpm (aka rotational speed), the more "strokes" of the Displacer …

hence a greater Capacity/Flowrate of pumped-up liquid will be discharged from the Reciprocating-Action PD Pump over a unit of time (for example, a minute).

In other words:

IF:

The Crankshaft in the nearby gif is moved back and forth more quickly  ...

THEN

There will be a greater Capacity/Flowrate of pumped-up liquid discharged from the pump over an interval of time.

Like, Duh!

The third "law" is not quite so intuitive:

The Capacity/Flowrate of the Pump is proportional to "the square of the diameter" of the Displacer.

Not to worry, Fred! The "law" can be easily explained with the below mathematical assessment which applies to all Single-Acting Reciprocating-Action Pumps.

 

THE MATHEMATICAL EXPRESSION FOR THE CAPACITY/FLOWRATE (AKA "Q") OF A SINGLE-ACTING RECIPROCATING-ACTION PUMP 

Once again, observe the movements of the nearby Single-Acting Reciprocating-Action PD Pump gif:

Given the Following Mathematical Definitions:

L (inches) = Stroke Length of the Displacer ... aka the forward distance that the Displacer "travels" on each Stroke.

N (Strokes/Minute) = Stroke Frequency of the Displacer (of course powered by the unseen Driver)

D (inches) = Diameter of the Displacer (a Piston, Plunger, or Diaphragm)

3.14 = Pi (aka Greek letter Π)

A (inches²) = Surface Area of the Displacer = [(3.14)* (D²) /4)]

Hey! Looky there!

Determining the Surface Area of the Displacer (A) requires "squaring the Diameter (D) of the Displacer!"

To make Law #3 make sense, the Surface Area of the Displacer (A) must be included in the mathematical definition of Capacity/Flowrate discharged from a Reciprocating-Action Positive-Displacement Pump!

 

The Mathematical Expression for Capacity/Flowrate Discharged from a Reciprocating-Action PD Pump

Q is the Capacity/Flowrate discharged from a Reciprocating-Action PD Pump.

Q is a "volumetric flowrate" meaning the units that describe Q will be a expression of "volume" divided by an expression of "time."

Q (inch³/min) = A * L * N

In English:

 

The Capacity/Flowrate discharged from a Reciprocating-Action PD Pump is determined by multiplying:

 

Surface Area of the Displacer * Stroke Length * Stroke Frequency

The Capacity/Flowrate expressed in gallons per minute … gpm … makes more sense to the human being. PTOA Readers and Students can employ the factor-labelling method featured in PTOA Segment #148 and multiply by 0.004329 to convert cubic inches/min to gpm because:

Q in³/min * 1 ft³/1728 in³ * 7.481 gal/1 ft³ = Q * 0.004329 gal/min or gpm

Uh oh! Fred is still mega worried and confusedl

Fred! Keep reading and you will be empowered to determine the Capacity/Flowrate from any Reciprocating-Action PD Pump!

For example:

 

Assume the Piston in the Single-Acting Reciprocating-Action PD Pump shown nearby:

• Has a 3 in Diameter (D).
• Travels 6 inches forward per each Stroke (L).
• The Driver rotates at 475 rpm, which means the Crankshaft moves the Piston forward 475 Strokes/min (N).

 

Therefore the Capacity/Flowrate discharged from this Pump can be determined:

Q gal/min = A * L * N * 0.004329 =

[(3.14)* (3 in²) /4)] * 6 in/Stroke * 475 Stroke/min * 0.004329 gal/in³ = 87.17 gpm 

Hey! That calculated answer states that the amount of liquid discharged from the described  Reciprocating-Action PD Pump is just above 87 gallons each minute!

The blue drum shown nearby can hold 55 gallons.  A Capacity/Flowrate of 87 gpm means 1 full drum plus 79% of a second drum would be needed to hold the amount of liquid discharged from the Pump every minute!

 

 

 

EFFICIENCY OF A RECIPROCATING-ACTION POSITIVE DISPLACEMENT PUMP,

Assuming the Driver power output remains unchanged …

The Efficiency of a Reciprocating-Action PD Pump will be increased (or decreased) anytime more (or less) Capacity/Flowrate (Q) is discharged with each Stroke of the Displacer.

Options for Increasing Capacity/Flowrate aka "Q" 

Recall that the "good things" about Reciprocating-Action PD Pumps include the efficiently-delivered, sustained-yet-moderate Capacity/Flowrate (Q) of a liquid that is discharged at whatever high PV Discharge Pressure the Driver and strength of the pump can endure.

What if even more Capacity/Flowrate (Q) is needed for the industrial process? What options are available without sacrificing the Reciprocating-Action PD Pump's Efficiency?

The Double-Acting Reciprocating-Action Pump featured in PTOA Segment #109 will discharge twice as much Capacity/Flowrate as a similarly-sized Single-Acting Reciprocating-Action PD Pump.

Therefore the Double-Acting Reciprocating-Action PD Pump is twice as Efficient as a similarly-sized Single-Acting Reciprocating-Action PD Pump.

The increase in Capacity/Flowrate (Q) can be calculated and thus quantified using the below table:

Q = (A * L * N)  for a Single-Acting Reciprocating-Action Pump

Q = 2 * (A * L * N) for a Double-Acting Reciprocating-Action Pump

Q = 3 * (A * L * N) for a Triplex Pump

Likewise, the Triplex Pump will discharge thrice as much Capacity/Flowrate as a Single-Acting Reciprocating-Action Pump with a similarly-sized Displacer Diameter, Stroke Length, and Driver speed.

Would some of the power from the Driver be lost in the Gear Box?  A modern short-stroke Reciprocating-Action PD Pump with a 300-720 rpm Driver speed might be directly connected to the Driver with no gear reduction. Some models might require a single-reduction gear drive.

 

CHANGING DISPLACER SIZE IMPACTS RECIPROCATING-ACTION PD PUMP EFFICIENCY 

To Recap:

Reciprocating-Action PD Pump Rules #1 and #2 predict that the Flowrate/Capacity (Q) will increase/decrease if the Driver speed increases/decreases the Strokes per Minute (N) of the Displacer.

Reciprocating-Action PD Pump Rule # 3 logically predicts that the Capacity/Flowrate (Q) of the pump will increase when additional Displacers with Surface Area (A) are engaged to travel back and forth a Distance (L) at a rate of (N) Strokes per Minute.

Even the Capacity/Flowrate (Q) of a Single-Action Reciprocating-Action PD Pump can be increased or decreased by changing the size of the Piston or Plunger.  Many Reciprocating-Action PD Pumps are fabricated so that the size of the Piston or Plunger can be changed easily.

 

How Does Changing Pump Internals Impact Reciprocating-Action PD Pump Efficiency? 

As stated above, the Efficiency of a Reciprocating-Action PD Pump is impacted any time it takes the same amount of power from the Driver to displace less Capacity/Flowrate from the pump.

Changing the Piston or Plunger to a smaller Diameter Piston or Plunger without changing the power input from the Driver will displace LESS  liquid from the Reciprocating-Action PD Pump. Thus the Efficiency of the pump will decrease.

Why would a Pump Mechanic intentionally change to a smaller diameter Displacer knowing that maintaining the Driver at the same speed will decrease the pump's Efficiency?

Maybe the change was necessary to gain more PV Discharge Pressure so that the Reciprocating-Action PD Pump (which has already been purchased and installed) can deliver the liquid where it needs to go!

Uh-oh Fred is confused. Don't worry Fred!  Just keep reading!

 

 

THE RELATIONSHIP BETWEEN CAPACITY/FLOWRATE (Q) AND PV DISCHARGE PRESSURE AT CONSTANT DRIVER POWER INPUT

What will happen to the PV Discharge Pressure from the Pump when the Capacity/Flowrate (Q) is increased or decreased?

WHEN THE POWER OUTPUT FROM THE DRIVER IS HELD CONSTANT …

 

The Capacity/Flowrate (Q) discharged from a Reciprocating-Action PD Pump will vary inversely with Discharge PV Pressure!

 

IF Q ↑,THEN PV DISCHARGE PRESSURE ↓

IF Q ↓, THEN  PV DISCHARGE PRESSURE ↑

The nearby graph shows the relationship between the Capacity/Flowrate (Q) of a PD Pump on the Y Axis compared to the Pump Speed in RPMs (powered by the Driver) on the X Axis.

The Pump speed in rpms is directly related to N (Strokes per Minute) of the Displacer.

A constant Pump Speed at 250 RPM is shown as a single dashed VERTICAL red line.  This line represents "constant power output from the Driver."

The four red, HORIZONTAL  dashed lines point to a range of Capacity/Flowrates (Q) on the Y axis ranging from roughly 338 gpm to 488 gpm.

Four slanted lines indicate the PV Discharge Pressure of the Pump. Each of these lines intersects one of the four Capacity/Flowrate-Pump Speed/RPM relationships depicted on the graph.  The PV Discharge Pressures range from 25 psi to 200 psi.

The conclusions from the graph are:

At a constant pump power output from the Driver at 250 rpm:

 

• A Capacity/Flowrate (Q) of approximately 338 gpm results in a PV Discharge Pressure of 200 psi.
• A Capacity/Flowrate (Q) of approximately 376 gpm results in a PV Discharge Pressure of 100 psi.
• A Capacity/Flowrate (Q)of approximately 413 gpm results in a PV Discharge Pressure of 50 psi.
• A Capacity/Flowrate (Q) of approximately 488 gpm results in a PV Discharge Pressure of 25 psi.

 

The graph clearly proves that … when the power input to the Reciprocating-Action PD Pump is held constant ...

The PV Discharge Pressure of the liquid being pumped up will increase when the Capacity/Flowrate (Q) decreases, and vice versa.

But why exactly is this outcome observed?

 

The Difference Between Hydrostatic and Hydrodynamic PV Pressure … One Last Time

PTOA Readers and Students learned in PTOA Segment #153 that liquids are not compressible but gases are.

Thus the two methods used to increase the PV Pressure in a liquid are:

hydroDYNAMICALLY (e.g., a Centrifugal Pump) or

hydroSTATICALLY (eg. a Positive Displacement Pump).

Who could blame any PTOA Reader or Student for blurting out that ye olde PV Pressure↔Fluid Velocity Swap must be the reason for the inverse relationship between the Capacity/Flowrate (Q) and the PV Discharge Pressure of a Reciprocating-Action PD Pump?

After all the PV Pressure↔Fluid Velocity Swap is the operating theory behind all Centrifugal Pumps (PTOA Segment #173) and some Special Dynamic Pumps (PTOA Segment #207) and some Artificial Lift technologies (PTOA Segment #208).

However, for the PV Pressure↔Fluid Velocity Swap to work, the liquid in question must be flowing so that it actually has some velocity to swap into the PV Pressure, and vice versa. The ten dollar word to describe a liquid that is flowing is to say that the liquid is in a "hydrodynamic state."

Positive Displacement Pumps contain liquids in Cylinders or other hardware where the liquid cannot "flow." The ten dollar word to describe a liquid that cannot flow is to say the liquid is in a "hydrostatic state."

Well, even though a hydrostatic liquid cannot flow it sure as heck can be "displaced."

PTOA Readers and Students learned all about hydraulic PV Pressure back in PTOA Segments #146 and #147).  Brilliant PTOA Readers and Students ... meaning those who are reading the PTOA Segments in the intended, sequential order ... know that each droplet of a contained liquid transfers the PV Pressure equally in all directions.

When the contained liquid is squashed into a smaller volume, an increase in PV Pressure is created because the same number of molecules of liquid are being squashed into a smaller container volume.  This increased PV Pressure is then transferred equally in all directions through the contained liquid.

With the Suction Valve closed and the Displacer pushing the mass of liquid into a smaller and smaller container, the PV Pressure of the liquid will continue to increase until it triggers the Discharge Valve to open at a preset PV Discharge Pressure setting.

To recap: There are two main ways to increase the PV Pressure in a liquid … hydrodynamically (ye old PV Pressure↔ Flud Velocity Swap) or hydrostatically (decreasing/displacing  the volume of a contained liquid).

Congratulations to all PTOA Readers and Students for finishing the PTOA Focus Study on Reciprocating-Action PD Pumps.

The next few PTOA Segments feature Rotary-Motion Positive Displacement Pumps.

TAKE HOME MESSAGES: The first two Reciprocating-Action PD Pump Rules state the direct relationship between an increase/decrease in Driver Speed and Strokes per Minute (N)  of the Displacer and the increase/decrease of the Capacity/Flowrate discharged from the pump. 

The third Reciprocating-Action PD Pump Rule states the direct relationship between the (squared) Displacer Diameter and the Capacity/Flowrate through the Pump; Increasing the Diameter of the Piston will significantly increase the Capacity/Flowrate … and vice versa.

For a Reciprocating-Action PD Pump, the Capacity/Flowrate (Q) can be calculated:

Q = (A * L * N)  for a Single-Acting Reciprocating-Action Pump. Otherwise stated:
Q = (Surface Area of the Displacer * Stroke Length * Stroke Frequency)

The third Reciprocating-Action PD Pump Rule can be extended to quantify the increase in Capacity/Flowrate which results when multiple Displacers are driven by the same power source.

Q = 2 * (A * L * N) for a Double-Acting Reciprocating-Action Pump

Q = 3 * (A * L * N) for a Triplex Pump

The Efficiency of a PD Pump is the amount of Capacity/Flowrate divided by the power input from the Driver.  Anytime the Capacity/Flowrate is decreased and the same amount or more power is required, the Efficiency of a Pump will be decreased … and vice versa. 

Some Reciprocating-Action PD Pumps are fabricated to make it possible to swap out Pistons or Plungers. When a smaller diameter Piston or Plunger is installed, a significantly smaller Capacity/Flowrate will be discharged from the pump. If the Driver speed is not increased, the Efficiency of the altered pump will be lower than the Efficiency of the original pump.  Such a change may still be warranted if more Discharge Pressure is needed.

When the power input from the Driver is held constant, there is an inverse relationship between Capacity/Flowrate and PV Discharge Pressure of the pumped-liquid. 

There are two main ways that the PV Pressure is added to liquids: 

Hydrodynamically (aka the PV Pressure↔Fluid Velocity Swap).

Hydrostatically (aka the basis of hydraulically moving liquids by displacement).

©2020 PTOA Segment 0210
PTOA PV PRESSURE FOCUS STUDY AREA
PTOA ROTATING EQUIPMENT AREA - DYNAMIC AND POSITIVE DISPLACEMENT PUMPS

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