A-to-Z Guide to Thermodynamics,
Heat & Mass Transfer, and Fluids Engineering
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Flow metering is the measurement of the quantity of a substance flowing through a given flow section in unit time. Flow metering is an unseparable part of any technological process flow chart; it is required to achieve maximum efficiency of automatization of production. It can also be widely used in scientific investigations.

The existing flowmeters can be classed into:

• Devices based on hydrodynamic methods: variable pressure differential, variable level, streamline flow, vortex methods and others;

• Devices with continuously moving bodies: tachometric, power, etc.;

• Devices based on various physical phenomena: thermal, electromagnetic, acoustic, optical, etc.

The devices of the first group have gained wide acceptance. Among them are the differential pressure flowmeters (see Differential Pressure Flowmeters), whose operation depends on the differential pressure across a device which changes the flow section of the pipe-line (Figure 1). The flowmeter includes: a primary transformer which creates a pressure differential; a differential pressure gauge which measures this pressure differential and connecting pipes. The most common type of primary transformers are restrictions (see Orifice meter): other types include a standard diaphragm, a standard nozzle, a pipe, Venturi nozzle, double diaphragms. Half-circle and quarter circle nozzles are used in measuring small flow rates.

Variable-level flowmeters are devices based on the dependence between the flow rate and the height of the level in the vessel into which a liquid is constantly supplied and, at the same time, flows out from the hole in the bottom or in a side wall. A vessel with an orifice which has an area S serves as a primary transformer. The quantity measured is the height of the level of a liquid z in the vessel, and the flow rate is defined from the relationship and does not depend on the density of the liquid (Figure 2), where α is the flow rate coefficient.

For measuring the flow rate of the liquid in open channels (trays) slit flowmeters are widely used which are miniature weirs in the wall of a vessel into which a liquid is constantly supplied. The flow rate is defined by the height of the level of a liquid above the lower edge of the slit. The characteristic of such a flowmeter depends on the shape of the profile of the slit cross section: for a rectangular slit b1 wide and b2 high, the flow rate is ; profiles b1(b2) have been developed on applying which a linear relationship is realized.

In streamline flow flowmeters, the primary transformer takes up the dynamic pressure of the flow and moves under its action by the value dependent on the flow rate. The most widely used are the constant pressure differential flowmeters in which a streamlined body moves vertically and a counterforce is produced by the weight of the body. These include Rotameters (Figure 3), and also floating and piston-type (slide valve) flowmeters. Rotameters are made in the form of a vertical conical (conicity, 0.001-0.01), upwardly divergent glass pipe, on which the graduations are made; a float, on whose upper rim inclined ribs are made, moves inside the pipe. The float comes upward and rotates under the action of the flow, the rotation ensures the centring of the float in the middle of the flow. The volume flow rate of the liquid is defined by the height z of the float lift. It depends on the float characteristics (its volume Vf, midsection area Sf = (π/4)df2 and on the density of the float material (ρf), the density p of the liquid measured and is proportional to the area of the circular gap Sc(z) between the walls of the pipe and the float, . For small taper angles of the pipe, the relationship Sc(z) is practically linear, in particular conditions of measurement , where A is defined by a preliminary calibration. The float-type flowmeters operate similarly. In piston-type flowmeters under the action of dynamic head the piston moves in a bush with specially shaped windows, through which the liquid flows out with a flow rate .

In vortex flowmeters, the frequency of variations of pressure or velocity which occur in the cross flow over a body (cylinder, prism, plate) (Figure 4) are determined and depend on flow rate. The frequency f is related to the mean streamlining velocity u and the size of the body d by the Strouhal number Sr = du−1f. For a flow section area S = (π/4)D2 the flow rate is defined by the relation . The proportionality between and f is ensured for Sr = const, which is realized when the cylinder is streamlined over the range 104 ≤ Ro ≤ 2 × 105 (Ro = ωd/u is the Rossby number). This ensures a wide range of the measured flow rate , but is limited by the conditions of steady vortex formation (for instance, for water u > 0.2 m/s). Primary transformers with d/D = 0.15 - 0.2 are usually used (D is the diameter of the pipeline). The pressure pulsations are transformed into an electric signal with the help of piezo ceramic pressure pickups. The error of flow rate measurement is estimated at 0.5-1.5%.

The tachometric flowmeters have a rotating element whose velocity of measurement is proportional to the volume flow rate.

Flowmeters in the form of a small turbine have found a wide utility, the speed of the turbine rotation being determined by the number of the electric pulses in unit time measured by a frequency meter (see Anemometers, Vane). Turbine flowmeters are designed either in the form of an axial small turbine (Figure 5) which has propeller blades with a variable helix angle, or in the form of a tangential small turbine (Figure 6) with flat radially arranged blades. In ball-type flowmeters a ball moving around the circuit due to the flow swirling with the help of a propeller guide is used as a moving element.

In power flowmeters, the value of a parameter is measured, which characterizes action of a force on the flow, the action being proportional to the mass flow rate. The flow is accelerated by the force. Depending on the character of acceleration, the flowmeters are classed as turbo-power (Figure 7) in which the flow is swirled either due to an external action (a rotor with an electric drive) or with the help of a fixed auger, Coriolis flowmeters in which Coriolis acceleration occurs due to the force action, and gyroscopic flowmeters in which the gyroscopic moment is measured. In the flowmeter shown in Figure 7, the torque on the rotating shaft is measured and is proportional to mass flow rate.

Thermal flowmeters (Figure 8) depend on the flow rate of the quantity of heat received by the flow of a liquid from a heater. A heater (usually an electric one) is placed into the flow on the pipeline section and its power W is measured, and the difference in the temperatures of the flow ΔT = Tout – Tin at the inlet and at the outlet. Then the mass flow rate is proportional to the power of heating W with ΔT kept constant. The coefficient K depends on heat losses into the surroundings, the nonuniform distribution of velocity along the pipeline cross section, etc., a preliminary calibration is therefore made. When the flowmeter is manufactured and calibrated carefully, it can give an accuracy of measurement of the flow rate of ±(0.3-0.5)% and can be used as a standard for testing and calibrating other flowmeters. The thermoanemometric method (see Hot-wire and Hot-film Anemometers) measures local velocity by determining the temperature attained by a hot wire or hot film fed with a constant current. This local velocity can then be related to the mean velocity using known relationships.

Electromagnetic Flowmeters (Figure 9) are designed to measure the flow rate of a liquid with conductance, as a rule, not less than 10−3 Ohm/m. Their operation relies on the interaction between the moving current-conducting liquid and the transverse magnetic field. In this case, an electromagnetic force E induced in a liquid, is proportional to the magnetic induction of the transverse field B, to the volume flow rate V of the liquid and to the distance D between the electrodes (located along the normal both to the velocity vector of the liquid and to the vector of the magnetic field intensity). D is typically equal to the inner diameter of the pipeline. The current generated (E) is given by . The electromagnetic flowmeter has the advantages of independence of viscosity and density of the substance, the absence of pressure loss, scale linearity, high speed response, the possibility of measiring in agressive, abraisive and highly viscous liquids. However, it cannot be applied for measuring flow rates of gas, vapors and dielectrics.

Acoustic (ultrasound) flowmeters (FLOW_METERING_FIG10) are based on measuring the difference in the time Δt for the passage of acoustic waves over a distance L in the direct of flow and counter to the direction of flow, respectively. The sound is emitted and received by the radiators detector devices R. The time Δt is proportional to L and to the mean mass velocity u and inversely proportional to the square of the velocity a of propagation of acoustic oscillations in the measured medium Δt = 2Lu/a2. Accounting for the fact that high-frequency oscillations (0.1 ... 10MHz)are applied for measuring very small time intervals, various methods have been developed. Most extensively employed is the Doppler shift method in which the difference of pulse frequency rate along and against the flow is measured. The volume flow rate can be defined from the relationship . The flowmeter consists of a pipe section in whose end faces disc piezoceramic (for instance, from barium titanium BaTiO3) elements are installed which are switched on alternately to radiation or to reception. Ultrasound flowmeters are particularly useful for measuring the flow rate of nonconducting liquids, of petroleum products and agressive media. The accuracy of measurement is estimated at 0.5-1%.

Optical flowmeters use the dependence on the flow rate of the substance of one or the other optical effect in the flow. The most generally employed are laser Doppler flowmeters (see Anemometers, Laser Doppler) which rely on measuring the difference of frequencies Δf of incident and reflected light, which is due to the interaction between the light beam and the particles moving in the liquid. In this case the local velocity of the flow in some point of the cross section of the pipe is measured; if the relationship between the local and mean velocity is known, the flow rate may be determined.

#### REFERENCES

Fluid Meters: Their Theory and Application (1971) New York, ASME.

#### References

1. Fluid Meters: Their Theory and Application (1971) New York, ASME.