Measurement of the flow rates of liquids, gases and vapors with orifice meters has found wide use both in industrial and in scientific measurements. A restriction fulfilling the function of a primary converter, is installed in a pipeline and produces in it a local change of a flow section. The method depends upon the fact that an increase in velocity and kinetic energy of the flow behind the restriction as compared with the parameters upstream of it, brings about a decrease in a static pressure p_{out} downstream of the restriction with respect to the pressure p_{in} upstream of it. The differential pressure Δp = p_{in} −> p_{out} depends on the fluid flow velocity and can serve as a flow rate measure (see Differential pressure flowmeter).

The following designs of orifice are used: the standard orifice (Figure 1a), the double orifice (Figure 1b), orifice with an input cone (Figure 1c), and an orifice with a double cone (Figure 1d). Closely related to orifices, but offering advantages of better downstream pressure recovery and (sometimes) range, in exchange for increased cost, are a whole variety of other restriction designs. These include standard nozzles (Figure 1e), "half-circle" nozzles (Figure 1g), "quarter-circle" nozzles (Figure 1i), cylindrical nozzles (Figure 1k), Venturi tube (Figure 1l), Venturi nozzle (Figure 1m), and Dall's tube (Figure 1n).

The most widely used standard restrictions are orifice plates converging nozzles and Venturi nozzles (see Figure 2). The orifice plate is a thin disk with a hole with diameter d and area S, located in line with the pipeline whose diameter is D. The nozzle is made in the form of an insert with an orifice smoothly contracting at the inlet and ending with a cylindrical part. The nozzle profile allows us to realize a smooth compression of a jet up to its minimum section, which ensures less overall loss of pressure than in the case of the orifice plate. The Venturi nozzle has minimum losses of pressure of all the restrictions due to the installation of a diffusor at the outlet, which recovers the pressure. The principle of measuring the substance flow rate of pressure differential is the same for all types of restrictions, and a quantitative relation between and Δp is defined by the relation

where ρ is the fluid density upstream of the meter and e is a correction factor for compressibility (to be discussed below). The coefficient of proportionality α is called the "flow rate coefficient" or "discharge coefficient" and depends on the restriction type, on the degree of jet contraction (i.e., on the ratio of the flow area of the restriction to the cross sectional area of the pipe, where m = d^{2}/D^{2}), and on the Reynolds number Re = ūD/v defined by the mean velocity ū of the fluid.

For a given value of area ratio (m) α varies significantly with Re for small Re, but for Reynolds numbers higher than a boundary value Re_{b} (Re > Re_{b}) the flow in the restriction is self-similar and α is independent of Re and is constant for the given type of restriction. In measurements at low Re, double orifice plates are used, where an additional orifice with m_{1} > m_{2} is installed upstream of the main orifice, and the pressure taps are located immediately upstream of the additional and downstream of the main plate. Nozzles with the "quarter-circle" profile are also useful in this zone.

The values of boundary Reynolds numbers (Re_{b}) are given in Table 1.

**Table 1. Values of boundary Reynolds number (Re**_{b} × 10^{−4})

The correction factor for the expansion of the medium ε being measured accounts for the changes in the substance density on its flowing through a restriction and depends on Δp/p_{in}, on the area ratio m, on the adiabatic exponent (γ) of the substance and on the type of the restriction. For standard diaphragms empirical equations can be used, for instance,

where δp = (1 — Δp/p_{in}). For standard nozzles and Venturi nozzles

For measuring the flow rate of a liquid ε = 1.

Suppose we wish to measure flow of a substance with density ρ flowing in a tube of diameter D. We wish to determine the flow rate up to a maximum value (kg/s) and the maximum pressure drop we wish to allow over the orifice is Δp_{max}(p_{a}). In order to determine the diameter d, we invoke the relation

The relation between ma and a for standard restrictions is given in Table 2.

Using α determined from the table, we can calculate m = (mα)/α and the diameter of the restriction: d = Dm^{0.5}. When metering hot fluids, it is sometimes important to take account of changes in orifice diameter as a result of thermal expansion; if the temperature of the operating medium flowing through the restriction is T and the diameter at 293 K is d, the diameter is found from the relationship d[1 + β(T – 293)], where β is the linear expansion coefficient of the material from which the restriction is manufactured.

**Table 2. Relation between mα and α for standard restrictions**

In order to carry out the precision measurements, the flow rate coefficient α is often initially determined from the results of calibration by the weight or by the volume method.

The density of a substance flowing through a restrictor is determined from the pressure p_{in} measured directly at the inlet end face of the restriction, through separate holes which are not used for measuring Δp and the temperature T_{in} is determined with the help of a thermometer located at a distance of l = (15–20)D from the front end face of the restriction.

The standard diaphragm is shown schematically in Figure 2a. The hole in the diaphragm must have a cylindrical form with a sharp edge at the inlet. The length of the cylindrical part is 0.005 ≤ l ≤ 0.02D, and for m > 0.5, l = b/3. The diaphragm thickness should be b ≤ 0.05D, but not less than 2.5–3 mm. At the outlet the hole is made conical with an angle of half-opening 30–45°. The relative area of the orifice m should be in the range 0.05–0.64. The inlet and outlet planes of the orifice plate must be parallel, and the angle between the inlet plane of the orifice plate and a normal to the tube axis must not exceed 1°. Special attention in manufacturing must be given to treatment of the inlet edge; it must be free of chamfering and scratches. Pressure must be tapped with the help of holes which must be drilled in the immediate vicinity of the front and rear walls of the orifice plate, or through the use of circular chambers connected with the inner cavity of the pipeline by an annular slot or by several holes. In this case the pressure is averaged over the slot perimeter. The diameter of the orifice or the slot width c ≤ 0.01 D, in doubled orifices it is 0.5D. At small Re, orifices with an inlet cone, with a double cone or with a rounded surface are used.

The standard nozzle is shown in Figure 2b. The nozzle consists of an inlet part formed by the arcs of the circles of radii r_{1} = 0.2d and r_{2} = 0.33d, which transfer smoothly from one to the other, and of a cylindrical part of length l = 0.3d at the outlet. The arc of radius r_{2} transfers to the cylindrical part along the tangent, and the arc of radii r_{1} is integrated with the inlet plane. The deviation of radiuses r_{1} and r_{2} from the nominal value must not exceed 10 percent. At the outlet, the cylindrical part of the nozzle orifice has a bore of diameter 1.06d and of length 0.3d to prevent the outlet edge from damages. The ellipticity of the nozzle orifice must not exceed 0.1 percent. Such nozzles have a coefficient of jet constriction close to unity, as a result of which the discharge coefficient is considerably higher than that of the orifice plate, which allows us to measure larger flow rates at the same differential pressure.

In the "quarter-circle" nozzle, the nozzle profile is formed by one arc, whose length is equal to 1/4 of the length of the circle of radius r, and a cylindrical part is absent. With the aim of ensuring constancy of the discharge coefficient a, the choice of the value of r is of fundamental importance. Table 3 gives recommended values.

**Table 3. Recommended radii for the quarter-circle nozzle**

Nozzles with "half-circle" profile, rounded profile and combined nozzles are also used.

A restriction which has a diffusor at the end for recovering pressure is called a flow tube. To this class belong the Venturi tube and the Venturi nozzle. The use of these devices is also described in the article on Venturi Meters. The Venturi tube consists of an inlet cylindrical part with diameter D and of length 1 = D; of a confusor with a taper angle of 21 ± 1° of length 2.7(D–d); a cylindrical throttle of length equal to its diameter d, and a diffusor with conicity of 7–8°, which has the diameter D at the outlet. The application of shortened Venturi's tubes with the outlet diameter less than D and the diffusor conicity of 14–16° with the diffusor length of (0.7–1) D is sometimes encountered.

The Venturi nozzle (Figure 2c) has an inlet part made according to the standard nozzle profile, but a somewhat lengthened cylindrical part (0.7–0.75d instead of 0.3d). The angle of flare of the diffusor is φ = 12–16°. The pressure pin is tapped immediately ahead of the inlet plane, and the pressure pout is tapped through four holes spaced around the circumference at a distance of 0.3d from the beginning of the cylindrical part. The application of shortened Venturi nozzles with a cut off diffusor is also encountered. Besides the above standardized geometries double Venturi nozzles and the Dall tube, which have low pressure losses, are also used. Of particular importance for increasing the accuracy of measurement of the flow rate are the conditions of mounting the restriction into the pipeline system. This defines the necessity of straight sections in a pipeline in front of the restriction, l_{in}, and behind it, l_{out} Depending on perturbations introduced into the flow by the pipeline elements, the minimum length of the straight part ahead of the restriction for orifice plates and nozzles is l_{in} = (60–80) D, and l_{out} = (4–8) D. For Venturi nozzles l_{in} = (3–4) D and l_{out} = (6–8) D. The required lengths l_{in} and l_{out} increase with increasing m.

The restriction is connected with the differential pressure gauge by two tubes. In standard flowmetering devices, the diameter of these tubes is 8–12 mm and the length of the tubes must be minimal wherever possible. The lines must have a slope not less than 1:10 along the entire path. The tube ends must be smooth, without acute angles and hollows. Steel, copper brass, and aluminum tubes are used. Tubes made from plastic materials can be used at pressures up to 0.6 MPa.

Differential pressure gauges are used for measuring pressure differential Δp in the restriction: U-shaped and cup-type manometers, inclined alcohol micromanometers, compensation micropres-sure gauges which allow the measurement of Δp with an accuracy of ±0.1 Pa, liquid-sealed bell manometers and ring-balance manometers, diaphragm pressure gauges, bellows pressure gauges, etc. (see Pressure Measurement). The accuracy of measuring the flow rate on the employment of standard restrictions is estimated as 0.8–2 percent for liquids and 1–3 percent for gases. On carrying out an individual calibration of the restrictions, the accuracy increases significantly.

#### REFERENCES

Fluid meters, their theory and application significantly (1971) Report of ASME, Research Committee on Fluid Meters, New York, published by ASME.