# 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.

Linear (mA) | Square root (mA) | Square root (mA) | Linear (mA) |

4 | 4.00 | 4 | 4.00 |

5 | 8.00 | 5 | 4.06 |

6 | 9.66 | 6 | 4.25 |

7 | 10.93 | 7 | 4.56 |

8 | 12.00 | 8 | 5.00 |

9 | 12.94 | 9 | 5.56 |

10 | 13.80 | 10 | 6.25 |

11 | 14.58 | 11 | 7.06 |

12 | 15.31 | 12 | 8.00 |

13 | 16.00 | 13 | 9.06 |

14 | 16.65 | 14 | 10.25 |

15 | 17.27 | 15 | 11.56 |

16 | 17.86 | 16 | 13.00 |

17 | 18.42 | 17 | 14.56 |

18 | 18.97 | 18 | 16.25 |

19 | 19.49 | 19 | 18.06 |

20 | 20.00 | 20 | 20.00 |