I can't live if living is without you
I can't live, I can't give anymore
I can't live if living is without you
I can't give, I can't give anymore

("Without You," P.Ham & T. Evans of Badfinger, 1971)

NO CHANGE IN PRESSURE?  THEN NO FLOW!

PTOA Readers and Students who are reading the PTOA Segments in the intended sequential order know that this PTOA Segment #158 is the second in a series that focuses on the interrelationship between the PV Pressure and another PV ... in this case the PV Flowrate.

Actually ...

There are two crucial relationships between the PV Pressure and the PV Flowrate that will be featured in this and the next PTOA Segments.

Guess what!

The PV Pressure - PV Flowrate relationship is totally one-way!

From the point of view of the PV Flowrate ...

the relationship looks like:

Because ...

The PV Flowrate cannot exist without the PV Pressure!

To be more specific, the PV Flowrate is dependent upon a Pressure Differential ...

which is interchangeably known as:

• Delta P
• ΔP
• dP
• Pressure Drop 
• Pressure Gradient

On the other hand ...

The PV Pressure can and does exist independently without creating the PV Flowrate.

For example,

PTOA Readers and Students already learned in PTOA Segments #146 and #147 that the PV Pressure exists anytime a pooled, homogenous liquid is stored in a container ... like a tank.

So, from the point of view of the PV Pressure, the relationship between the PV Pressure and the PV Flowrate is more like the relationship shown in the nearby photo.

The purpose of PTOA Segment #158 is to clarify the one way relationship between a change in the PV Pressure (aka Pressure Differential) and the PV Flowrate.

 

QUICKIE REVIEW OF "TRANSPORT PHENOMENA" & THEIR DRIVING FORCES

PTOA Readers and Students were first introduced to the cosmic sounding "Transport Phenomena" that rule the Universe way back in PTOA Segment #57.

"Transport Phenomena" just means that:

• Something ... like Heat or Current or Fluid ... is flowing from one place to the other... and
• The rate at which that Something is flowing depends upon a driving force that is more concentrated in one area than in another area ... and
• The Something always moves from the area of high concentration to the area of lower concentration.

 

Quickie Review of the Heat Transfer and Delta Temperature (ΔT)

When it comes to HEAT that is being "transported" ... aka transferred...

PTOA Readers and Students already expertly understand that HEAT flows from a warmer area to a colder area ...

and that it is the difference in Temperature between the two areas (ΔT) which provides the driving force that automatically results in HEAT being transferred ... or moved ... between the two areas.

Furthermore, the greater the ΔT, the greater the amount of heat that will flow between the two areas over a unit of time ...

until the entire area is at the same final Temperature!

So the driving force for HEAT TRANSFER is the ΔT between the hotter area and the colder area.

PTOA Readers and Students learned all about Heat Transfer in PTOA Segment #58, one of the first focus studies in the PTOA Heat Transfer Focus Study series.

So PTOA Readers and Students who are reading the PTOA Segments in the intended sequential order already thoroughly understand how and why a ΔT provides the driving force for the three methods of HEAT TRANSFER: conduction, convection, and radiation.

 

Quickie Review of Electrical Current and Delta Voltage (ΔV).

PTOA Readers and Students also learned that a difference in Voltage (ΔV) is needed before electrons can flow around a circuit.

 

ΔV is often called a "Potential Difference" or even more simply "Potential."

Electrons will always flow from the area of highest voltage (usually supplied by a battery) to an area of less voltage.

Furthermore the greater the ΔV, the greater the amount of current that flows around a circuit.

The relationship between current and ΔV was introduced in PTOA Segment #57 and further explored during PTOA Segment #106 which introduced electrical instruments that measure the PV Temperature.

 

FLUID FLOW DEPENDS UPON DELTA P (ΔP)

The last "Transport Phenomena" that has many applications in the process industries is the dependence that the PV Flowrate has upon the existence of a Pressure Differential.

Without a Pressure Differential, a fluid Flowrate cannot exist!

PTOA Readers and Students learned a long time ago in  PTOA Segment #3 that the term "fluid" applies to both liquids and gas/vapors.

In the much more recent PTOA Segment #140, PTOA Readers and Students learned that  ... under certain conditions ...

even solids like catalysts can be made to "fluidize."

In all cases ...

getting a fluidized solid or a liquid or a gas/vapor to flow requires a Pressure Differential.

Beware and always remember!

The term "Pressure Differential" can interchangeably be called:

• Delta P
• dp
• ΔP
• Pressure Drop
• Pressure Gradient

The other parts of the "Transport Phenomena" also apply to the  ΔP - Flowrate Relationship:

• The direction of the Flowrate is always from the area of High Pressure to the area of Low Pressure.
• The greater the ΔP, the greater the PV Flowrate over a unit of time.

 

MEASURING FLUID FLOWRATES WITH LIQUID HEAD INSTRUMENTS

As all PTOA Readers and Students would expect ...

Mankind has devoted hours observing the relationship between ΔP and Flowrate to whittle it down into a mathematical expression.

The complexity of the final version of the mathematical expression exceeds the purview of the PTOA and is not a requirement for Process Operators to know.

However,

the simplified expression shown below applies when the PV Flowrate is measured via intentionally generating a ΔP from a head of fluid:

PTOA Readers and Students will learn about Flowrate measuring instruments in the PTOA PV Flowrate Focus Study Area that follows the PTOA PV Pressure Focus Study Area.

The terms in the above expression are:

Q ... the VOLUMETRIC FLOWRATE that in the real world is expressed with units like "cubic meters per hour" (M³/hr) or "gallons per hour" (gal/hr) or "cubic feet per hour" (ft³/hr) or "barrels per day" (Bbls/day), etc.

Note that Q is not the same Q that was used in HEAT TRANSFER equations for Btu/hr ... that was a different Q!

K is, of course... another "fudge factor" that helps the units on both sides of the equation work out as well as takes into account stuff like the density of the fluid that is flowing and the diameter of the pipe that the fluid is flowing through.

ΔP is the Pressure Differential (aka Delta P aka dP aka Pressure Drop aka Pressure Gradient) created between taps that are inserted to sense an area of High Pressure and the area of Low Pressure.

Hey!

There's something kind of weird about the  ΔP - Flowrate expression above so let's scrutinize it again:

Aha!

The expression clearly shows that the amount of Flowrate (Q) is proportional ...but NOT LINEAR ... to the ΔP that is generated by the head-producing instrument.

Unlike the flow of Heat and its driving force, ΔT  ...

Unlike the flow of Current and its driving force, ΔV ...

Flowrate is proportional to the SQUARE ROOT of the ΔP !

To get ΔP out from under the square root sign, both sides of the equation must be squared ... that's just how math works!

That makes ΔP equal to the square of the Flowrate (Q²)!

The below graphic shows that version of the ΔP - Flowrate relationship ...

That weird little symbol between both sides of the ΔP - Flowrate expression means "approximates" ...

because once the fudge factor K is taken out of the expression the best one can do is an approximation!

The graphic below is just like the graphic above except that it has a real Flowrate expressed as gallons per minute (gpm).

The ΔP is in units of pounds per square inch (psi).

Your Mentor demonstrated how to convert a liquid head into a psi pressure in PTOA Segment #149.

Nowadays modern Process Operators are not consciously aware that the ΔP - Flowrate relationship is not linear!

Your Mentor is old enough to remember when Flowrates were displayed on square-root chart paper.

But today's modern DCS systems automatically apply a "square root extraction function/algorithm" which  modifies the Flowrate standard signal that is transmitted into the modern Man-Machine Interface.

So this linearizing algorithm takes the real-world relationship shown on the left side of the graphic below and straightens it into a one-to-one linear correspondence between the square root of ΔP and Flowrate ... as shown in the right side of the graphic below:

So Your Mentor can't really fault modern Process Operators who never knew or don't remember that when Flowrate is inferred from a Head Instrument ... the Flowrate is proportional to the SQUARE ROOT OF ΔP.

However, Instrumentation Technicians must never forget the special relationship between ΔP - Flowrate because they frequently calibrate flowrate instruments that intentionally generate a liquid head Pressure specifically for the purpose of inferring a Flowrate measurement.

And one thing all of the ΔP - Flowrate graphics above show ...

corrected for square root function or not ...

When the ΔP goes to zero ... so does the Flowrate!

 

DIY ΔP - FLOWRATE  EXERCISE:

So let's see which of the smart PTOA Readers and Students got the main points of the ΔP - Flowrate relationship by answering the following questions:

You are the Outside Process Operator doing rounds checking the local TIs, PIs, and FIs (Flow Indicators).

While checking the below long pipeline you notice that a Pressure Indicator (PI) indicates 150 psi (1034 kPa).

You walk down 200 more feet and notice the next PI that is sensing the pipeline Pressure also indicates 150 psi (1034 kPa).

Is there flow going through this pipe?

If so, in what direction is the flow going?

The answer will be in the next PTOA Segment.

 

TAKE HOME MESSAGES: The ΔP - Flowrate relationship is the third of three Transport Phenomena that govern how Heat, Current, and now Fluid flow in the Universe ... and therefore also in the process industries.

Flowrate is also known as "Volumetric Flowrate" and has units of Volume divided by time. For example:

• cubic meters per hour (M³/hr) ... fluid flow rate. 
• gallons per hour (gal/hr) or gpm (gal/min) ... liquid flow rates.
• cubic feet per hour (ft³/hr) ... a gas flowrate.
• Barrels per day (Bbls/day) ... a liquid flow rate.

The PV Flowrate cannot exist without a Pressure Differential. Otherwise stated, ΔP is the driving force that supports all fluid flow.

The term Pressure Differential can also appear as:

• Delta P
• ΔP 
• dp
• Pressure Drop
• Pressure Gradient

The Flowrate always flows from the area of higher pressure to the area of lower pressure.

The greater the Pressure Differential, the greater the flowrate.

When liquid head Pressure instruments are used to measure Flowrate, the Flowrate is proportional to the square root of the ΔP that is created.

Nowadays, the transmitters which transmit Flowrate measurements as standard signals apply an algorithm called a "square root extractor" that linearizes the Flowrate reading into a one-to-one correspondence with the square root of ΔP.

©2017 PTOA Segment 0158
PTOA Process Variable Pressure Focus Study Area
PTOA PV Pressure Interrelationship with PV Flowrate

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