Why square root is used in flow measurement?

• Differential pressure measurement is a commonly used method to measure the flow across an obstruction.
• Differential pressure is in direct proportion to the square of the flow.
• The square root relationship is applied only for DP-type flow measurements.
• These DP sensors or DP transmitters can measure differential pressure across various obstructions made in the flow channel or in a flow line.

• This differential pressure is developed by the use of primary elements such as an orifice plate, venturi tube, etc.
• But, differential pressure is not proportional to the flow rate across the orifice.
• The above relation is not a linear relationship or it is called the square root relation because the differential pressure measured has nonlinear relation with flow through the pipeline.
• The graphical relationship between differential pressure (ΔP) & flow rate (Q) is the quadratic parabola.

• A miserable significance of this quadratic relationship is that a flow rate is not directly sensed when a pressure-sensing instrument is connected to a flow sensor. But the pressure sensing instrument will sense the square of the flow rate.
• The square root function should be implemented in the loop for accurate measurement of flow throughout the operating range.
• The square root extractor converts the measured differential pressure into the rate of flow.
• Almost 50% of all flow-measuring applications use the differential pressure method to derive the flow rate. The relation between Differential Pressure (DP) and flow rate is given as

Where

Q = flow rate.

DP = differential pressure.

K = is a constant that depends on various factors such as the diameter of the pipe, orifice diameter, and so on.

What is square root extraction in flow measurement?

• A square root extractor is a type of signal conditioner.
• It is easy to use indicators, recorders, and controllers to register linearly with the flow velocity.
• Square root extraction in flow measurement is an arithmetic conversion applied to a linear measurement scale to convert it into a nonlinear square root scale.
• The square root scale is generated by considering the square root of the ratio between measurement reading and full span.
• Square Root Extraction is sometimes called square root scaling. And it is a fundamental step in considering the flow rate from a differential pressure measurement.

What is a square root extractor?

• A square root extractor converts a linear differential pressure signal to a nonlinear flow rate.
• The square root extractor is an electronic or pneumatic instrument that generates a corresponding linear flow signal from the flow transmitter.
• Usually, the flow transmitter signal is in the form of a square root.

Why flow rate is directly proportional to the square root of δp?

• According to Bernoulli’s principle “the square root of DP is directly proportional flow rate. It means if the pressure is high, then the flow rate increases and vice versa.
• The flow rate in the pipe is equal to the velocity multiplied by the cross-section because the pressure comes from only one end of the pipe.
• Flow element creates a pressure change by accelerating a fluid stream in differential pressure based flow meter.
• The pipeline is intentionally narrowed to create a low-pressure region.

Why square root is used in the flow transmitters?

Generally, the flow transmitter is configured in square root because a small change in differential pressure on the lower end results in a large change in flow.

If the square root is considered in the controller, the large change in flow makes it extra sensitive to electrical noise on the mA signal.

Where is square root extraction performed?

• Almost modern smart differential pressure transmitters allow the DP to square root conversion to be calculated within the transmitter.
• But, all SCADA and DCS systems have inbuilt square root extraction function blocks to configure inputs from DP transmitters.
• All square root extractions are performed in one or other of these locations.
• So, the instrument and control system engineer has an option of where to have the signal conversion.

Square root extraction formula:

A. To convert a given linear 4 to 20mA signal to a square root signal the following relation is adopted.

B. And to convert a square root signal to a linear 4 to 20mA signal use this formula:

Square root extraction table:

The table shows values for both a linear to square root conversion and a square root to linear conversion.