Square Root Extraction Method
Square root extraction, or square root scaling as it is sometimes called, is a fundamental step in inferring flow rate from differential pressure measurement. Below, we will demistify all that surrounds the daunting title - Square Root Extraction.
What is square root extraction?
Square Root Extraction is an arithmetical conversion which is applied to a linear measurement scale to convert it to a non-linear square root scale. For example, a square root extractor is used to convert a linear differential pressure signal to a non linear flow rate.
Why use a square root extractor?
The flow rate along a closed pipe is directly proportional to the square root of the pressure drop across a measuring device e.g. orifice plate, in the pipe, and it is estimated that around 50% of all flow measuring applications still use differential pressure to infer flow rate.
So, in these situations, square root extraction needs to be applied to the differential pressure reading to allow the flow rate to be inferred.
Where is square root extraction performed?
Most, if not all, modern smart differential pressure transmitters allow the dP to square root conversion to be calculated within the transmitter. All SCADA and distributed control systems have inbuilt square root extraction function blocks that can be configured for inputs from dP transmitters.
So, the instrument engineer (or perhaps the control system engineer) has the choice of where to have the signal conversion take place. Having said that, it is common practice to find a plant wide directive that all square root extraction will be performed in one or other of these locations. This avoids confusion, and the potential for error by converting a signal twice. If you do have a choice, ensure you document where the extraction is performed.
It is worth noting that it is not only modern smart transmitters that can perform square root extraction. For years, pneumatic square root extractors were common place in many process plants. They are still availabe for purchase from companies like Foxboro, Taylor etc.
Compensating for gas compressibility
When measuring gas flows it is not unusual to perform pressure and/or temperature compensation to the dP reading. In this case, the compensation calculations should be applied before the square extraction is performed. This would lead to the extraction being performed in the control system, unless the temperature and presure signals are availabe within the dP transmitter.
Square root extraction formula
To convert a a linear 4-20mA signal to a square root signal use the following formula:
[Output Sq Rt] = 4mA + (4 x Square Root of ([Output Linear] – 4mA))
And to convert a square root signal to a linear 4-20mA signal use this formula:
[Output Linear] = 4mA + (([Output Sq Rt] - 4mA)^2 / 16)
Square root extraction table
The table below depicts the major values for both a linear to square root conversion, and a square root to linear conversion. Values rounded as appropriate.
Technical Library
The following pages on Control and Instrumentation.com give more detail on flow measurement basics:
