## Flow meter composition and flow measurement principle

### Rotameter

The rotameter is composed of a tapered tube with a scale gradually expanding from bottom to top and a rotor placed in the tapered tube that can move up and down freely During operation, the measured fluid enters from the lower end of the tapered tube, moves up along the tapered tube, flows through the annular gap between the rotor and the tapered tube, and then flows out from the upper end of the tapered tube Driven by the flowing fluid, the rotor is subjected to the dynamic pressure of a bottom-up fluid on the rotor, which is exactly equal to the gravity of the rotor in the measured fluid (that is, the gravity of the rotor itself minus the buoyancy of the fluid on the rotor).

When the flowmeter is installed vertically, the center of gravity of the rotor is on the central axis of the cone tube, and the three forces on the rotor are parallel to the central axis. When the forces are balanced, the rotor is stabilized at a certain position in the cone tube. For a given rotameter, the material, size, and shape of the rotor can be determined, so the gravity of the rotor in the measured fluid is known, Only the dynamic pressure of the fluid on the rotor changes with the fluid velocity.

Therefore, when the fluid velocity increases or decreases, the dynamic pressure on the rotor increases or decreases, the rotor will move up or down, and the annular gap area between the rotor and the tapered pipe wall will also change, that is, the flow sectional area will also change. When the force on the rotor is balanced at a certain velocity, the rotor will be stabilized at a new position, For a given rotameter, the balance position of the rotor in the cone tube reflects the flow rate of the measured fluid through the cone tube.

### Differential pressure flowmeter

The differential pressure flowmeter consists of three parts, namely, a throttling device, impulse pipe, and differential pressure gauge The differential pressure flowmeter uses the throttling principle of fluid flow to realize flow measurement. The throttling principle is the phenomenon that when the fluid flows in the pipe with the throttling device, the static pressure of the fluid varies at the pipe wall before and after the throttling device.

There are two forms of energy in flowing fluid: static pressure energy and kinetic energy. Fluid has static pressure energy because of pressure and kinetic energy because of flow speed. Under certain conditions, these two forms of energy can be transformed into each other. According to the law of conservation of energy.

On the premise that the total energy of the static pressure energy and kinetic energy possessed by the fluid, plus the energy loss used to overcome the fluid flow resistance, is equal. Figure 2 shows the distribution of fluid pressure and velocity at sections I, II, and III before and after the throttling device Before reaching section I, the fluid flows at a certain velocity v1, and the static pressure is p1. When approaching the throttling device, the fluid near the pipe wall is blocked by the throttling device due to the obstruction of the throttling device, so that part of the kinetic energy is converted into static pressure energy, and the hydrostatic pressure of the fluid near the pipe wall at the inlet end of the throttling device increases, and is far greater than the pressure at the center of the pipe diameter, Therefore, a radial pressure difference is generated at the inlet end face of the throttling device.

Under the effect of the radial pressure difference, the fluid generates radial acceleration, so that the flow direction of the fluid particle near the pipe wall inclines to the central axis of the pipe, causing the phenomenon of pulse contraction. Due to the inertial effect, the minimum section of the flow beam is not at the orifice of the throttling device but continues to shrink after passing the throttling device, and the flow beam reaches the minimum at section II. At this time, the velocity is the maximum, that is, v2, and then the flow beam gradually expands, and completely recovers after reaching section III, and the velocity gradually decreases to the original value, that is, v3=v1.

Due to the local contraction of the flow beam produced by the throttling device, the flow rate of the fluid changes accordingly, that is, the kinetic energy also changes. According to the law of conservation of energy, the static pressure representing the hydrostatic energy of the fluid also changes. In section I, the fluid has a static pressure of p1 At section II, when the flow rate increases to the maximum v2, the static pressure decreases to the minimum p2 and then recovers with the recovery of the flow beam.

As the flow surface at the end face of the throttling device shrinks suddenly, and the flow area after the throttling device expands suddenly, the fluid forms a local eddy current, and part of the energy is consumed. At the same time, when the fluid flows through the orifice plate, it also needs to consume energy to overcome friction, so the static pressure p3 of the fluid at section III cannot be restored to the original value p1, resulting in permanent pressure loss Differential pressure at section I and II（ δ P=p1 - p2) has a one-to-one correspondence with the flow of fluid in front of the throttling device. As long as the pressure difference before and after the throttling device is measured, the flow can be expressed.

### Summary

The working principles of rotameter and differential pressure flowmeter are different The rotor flowmeter reflects the flow rate with the change of differential pressure under the condition that the throttling area (such as the flow area of the orifice plate) is unchanged; The differential pressure flowmeter, however, measures the flow by changing the throttling area with the pressure drop unchanged. That is, the measuring principle of the rotameter can be simplified as constant pressure drop, variable throttling; The measuring principle of differential pressure flowmeter is simplified as variable pressure drop, constant throttling.

## Flow equation derivation

### Rotameter

In the rotameter, when the rotor is stable, analyze the force on the rotor:

Including ρ T is the density of the rotor; ρ F is the density of the fluid; V is the volume of the rotor; Δ P is the pressure difference (constant) before and after the rotor; A is the maximum sectional area of the rotor.

The annular space area between the rotor and the tapered pipe is equivalent to the throttling area of the throttling flowmeter, but it is variable and has an approximately linear relationship with the rotor height h. Therefore, the flow of the rotameter can be expressed as:

Where ф Is the instrument constant; H is the rotor floating height

When the rotameter is calibrated in production, it is usually calibrated with water or air under the industrial reference state (20 ℃, 0.10133 Mpa) Therefore, in actual use, if the density and working state of the measured medium are inconsistent with the calibration, the flow indication value must be corrected according to the specific conditions of the density, temperature, pressure and other parameters of the actually measured medium.① Correction of liquid flow measurement

Since the rotameter for measuring liquid is calibrated with water at a room temperature of 20 ℃, it can be written as follows according to formula (1):Where qv0 is the flow scale when calibrated with water; ρ W is the density of water.

If the measured medium is not water, the flow scale needs to be corrected again. If the viscosity of the measured medium and the viscosity of water have little difference, it can be approximately considered as ф Is a constant, with

After dividing Formula (5) and Formula (4), we can get:

② Correction in gas flow measurement

When the rotameter is used for gas flow measurement, the flow value shall also be corrected. In addition to the density of the measured medium, the working temperature and pressure of the measured medium shall also be corrected When the display scale of the instrument is known to be qv0, the actual flow of the measured medium (industrial reference state) can be corrected according to the following formula:

Where, qvf is the actual flow of the measured medium; ρ 0 and ρ F is the density of air and measured medium under the standard state; Pf and Tf are the absolute pressure and thermodynamic temperature of the measured medium respectively; P0 and T0 are the absolute pressure and thermodynamic temperature under the standard state respectively (P0=0.10133 Mpa, T0=293K); Qv0 is the scale flow value.

### Differential pressure flowmeter

When the fluid flows through the throttling device, there is no external work, no external energy, and no temperature change of the fluid itself. For the fluid flowing in the pipeline, any two sections in the pipeline conform to the Bernoulli equation, and the selected sections I and II (see Figure 2) are analyzed.

Bernoulli equation of fluid:

According to equations (8) and (9), the flow can be deduced as:

α It is called the flow coefficient, which is related to such factors as the structure of the throttling device, the way of taking pressure, the ratio of orifice sectional area to pipe sectional area, Reynolds number, etc.

When equation (11) is brought into equation (10), the flow value can be obtained as:

For the flow monitoring of compressible fluid, the expansion coefficient should be introduced into the flow equation because it is prone to volume change ε， Then the basic flow equation can be written as:

Equations (13) and (14) are the flow equations of the throttling flowmeter, that is, the quantitative relationship between differential pressure and flow.

It can be seen from the basic flow equation that the flow is proportional to the square root of the pressure difference under other conditions. To know the true relationship between the flow and the pressure difference, the key is the α Value of α It is a comprehensive coefficient affected by many factors. For standard throttling devices, its value can be found in relevant manuals; For non-standard throttling devices, the values are obtained mainly by experimental methods.

### Summary

The two flow meters are based on different principles, and the flow equations obtained are quite different The basic flow equation of rotameter is derived from the balance of rotor force, while the basic flow equation of differential pressure flowmeter is derived from the Bernoulli equation and fluid continuity equation.

## Characteristics of flowmeter

### Rotameter

A rotor flowmeter is used to measure the flow of single-phase non-pulsating fluid (liquid or gas), which is widely used in chemical, petroleum, light industry, medicine, environmental protection, food, measurement and testing, scientific research, and other departments#### Advantages of rotameter:

① The rotameter is suitable for small pipe diameters and low flow rates. The diameter of the commonly used rotameter is below 40-50 mm, and the minimum diameter can reach 1.5-4 mm. When measuring the liquid flow rate, the flow rate of the glass tube rotameter with a diameter below 10 mm is only 0.2-0.6 m/s or even less than 0.1 m/s; The metal tube rotameter and the glass tube rotameter with a diameter greater than 15 mm have a flow rate of 0.5-1.5 m/s.

② The rotameter can be used at a lower Reynolds number. As long as the Reynolds number of the fluid flowing in the annular gap between the rotor and the pipe wall is greater than 40 or 500, the flow coefficient must remain constant even if the Reynolds number changes, that is, the fluid viscosity has no effect on the flow coefficient. This value is far lower than the throttling difference. Requirements for minimum Reynolds number of pressure instrument 104-105.

③ Most rotameters have no requirements for upstream straight pipe section, and have low requirements for installation conditions.

④ The flow measurement range of rotameter is wide, generally 10:1, the lowest is 5:1, and the highest is 25:1.

⑤ Compared with throttling flowmeter, rotameter has a lower pressure loss.

⑥ The glass tube rotameter has a simple structure, low price, and convenient use.#### Disadvantages of rotameter:

① If the fluid to be detected by the rotameter is different from the fluid used in factory calibration, the flow indication shall be corrected. The rotameter for measuring liquid is usually calibrated with water, and the gas is calibrated with air. If the density and viscosity of the fluid actually used are different, the flow shall deviate from the original graduation value, and conversion correction shall be made Therefore, the measurement accuracy is affected by the change of fluid physical parameters.

② The glass rotameter has a risk of being fragile due to its glass tube, especially the unguided rotor used to detect gas flow.

③ Most structure rotameters can only be used for pipe installation with vertical flow from bottom to top.④ The application of rotameter is only applicable to medium and small pipe diameters. The general full-flow rotameter is not applicable to large pipe diameters. The maximum diameter of a glass tube rotameter is 150 mm, and that of a metal rotameter is 200 mm.

### Differential pressure flowmeter

The differential pressure flowmeter is widely used and has a long history, and its usage occupies the first place among all kinds of flow meters Recently, the appearance of various new flowmeters has led to a decrease in its use. However, differential pressure flowmeters still play a decisive role in the whole flow measurement field and are widely used in petroleum, chemical industry, metallurgy, electric power, light industry, and other departments.

#### Advantages of differential pressure flowmeter:

① The standard differential pressure flowmeter is widely used, with a simple and firm structure, stable and reliable performance, long service life, convenient installation, and suitable for large flow measurement.② The standard throttling device is applicable to measuring pipes with diameters greater than 50 mm and Reynolds numbers above 104-105. The fluid should be clean and full of all pipes without phase change.

#### Disadvantages of differential pressure flowmeter:

① The measurement accuracy of differential pressure flowmeter is low, and the repeatability and accuracy of measurement are at a medium level among flowmeters. Due to the comprehensive influence of various factors, it is difficult to improve its accuracy.

② The flow measurement range is narrow. Since the flow is proportional to the square root of the instrument signal (differential pressure), the range is generally only 3:1-4:1③ The requirements for on-site installation conditions are relatively high. In order to ensure the stable flow of fluid before and after the throttling device, straight pipe sections (orifice plates and nozzles) of a certain length must be configured upstream and downstream of the throttling device, which is generally difficult to meet.

④ The differential pressure flowmeter has a large pressure loss, the orifice flowmeter has the largest pressure loss, the nozzle flowmeter takes the second place, and the venturi flowmeter is the smallest. It should not be used when large pipeline pressure loss is not allowed.⑤ The impulse pipeline between the test piece and the differential pressure display instrument is prone to leakage, blockage, freezing, signal distortion, and other faults.

### Summary

The differential pressure flowmeter is only suitable for measuring the fluid with a pipe diameter greater than 50 mm and a Reynolds number above 104-105, while the rotameter is suitable for measuring the flow rate with small pipe diameter, low flow rate, and low Reynolds number For differential pressure flowmeter (orifice plate and nozzle), in order to ensure the steady flow of fluid before and after the throttling device, straight pipe sections of a certain length must be configured at the upstream and downstream of the throttling device. For the rotameter, the requirements for upstream straight pipe sections are not high, and the on-site installation conditions are low The differential pressure flowmeter has a large pressure loss, while the rotameter has a low-pressure loss.### Conclusion

From the analysis of the working principle of the rotameter and the differential pressure flowmeter, the derivation of the basic flow equation, and the analysis of their advantages and disadvantages, the following understandings are obtained:The rotameter is a flow instrument with constant pressure drop and variable throttling area. Before leaving the factory, the rotameter is calibrated with water or air under the industrial reference state (20 ℃, 0.10133 Mpa). The basic flow equation needs to be corrected when it is used. It is applicable to small pipe diameters, low flow rates,s and low Reynolds numbers. The pressure loss is small.

The differential pressure flowmeter is a flow instrument with constant throttling and variable pressure drop. From the basic flow equation, it can be seen that the flow is proportional to the square root of the pressure difference on the premise that the flow coefficient, expansion coefficient, and throttling area remain unchanged. The pressure gauge is widely used, simple and firm in structure, stable and reliable in performance, long in service life, convenient in installation and suitable for large flow measurement with large pressure loss.When used in chemical production, the flowmeter shall be selected according to the site requirements and the characteristics of each instrument.