Expansion factor Y

When a fluid flows through a meter, there is a pressure drop as it passes through the constriction. When a compressible fluid flows through a meter, the resulting pressure drop causes a change in fluid density at the constriction. As a result, the fluid densities at the meter inlet and within the meter are different. The expansion factor corrects for density differences between pressure taps due to expansion to the lower pressure. 

The expansion factor for a compressible fluid can be estimated using the following equations  

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For the flow of gases, expansion factor Y, which allows for the change in gas density as it expands adiabaticaUy from pi to po, is given by... 

For the flow of liquids, expansion factor Y is unity. The change in potential energy in tne case of an inclined or vertical venturi meter must be allowed for. Equation (10-20) is accordingly modified to give... 

FIG. 10-16 Values of expansion factor Y for orifices, nozzles, and venturis. 

Rate of discbarge through an orifice meter is given by Eq. (10-8) for either hquids or gases. For the case of subsonic flow of a gas (/ < / < 1.0), the expansion factor Y for orifices is approximated by... 

For flow of compressible fluids use the net expansion factor Y (see later discussion) [3] ... 

Figure 2-38A. Net expansion factor, Y, for compressible flow through pipe to a larger flow area. By permission, Crane Co., Technical Paper U410, Engineering Div., 1957. Also see 1976 edition.

It is more convenient to express this result in terms of the ratio of Eq. (10-15) to the corresponding incompressible equation, Eq. (10-14), which is called the expansion factor Y ... 

The effect of variable density can be accounted for by an expansion factor Y as has been done for flow in pipes and meters, in which case Eq. (10-44) can be written... 

The expansion factor Y depends on the pressure drop X, the dimensions (clearance) in the valve, the gas specific heat ratio k, and the Reynolds number (the effect of which is often negligible). It has been found from measurements (Hutchison, 1971) that the expansion factor for a given valve can be represented, to within about 2%, by the expression... 

Values of y for supercritical flow of a gas (r < through orifices are given by Benedict [/. Basic Eng., 93, 121-137 (1971)]- For the case of liquids, expansion factor y is unity, and Eq. (10-27) should be used, since it allows for any difference in elevation between the upstream and downstream taps. 

Inasmuch as this equation is too complex for convenient use in metering, it is replaced by the simpler equation for an incompressible fluid, but with the insertion of an expansion factor Y. 

If the fluid is compressible, the specific weight will decrease from uq to w2 as the pressure drops from pi to p2 and the value of w determined by Eq. (10.89) will be larger than the true value W as given by Eq. (10.88). Therefore, we use an expansion factor Y such that W = YW. Therefore, the true flow rate for a compressible fluid is... 

Determine the expansion factor and the meter area factor. Since steam is a compressible fluid, the expansion factor Y must be determined. For superheated steam, the ratio of the specific heat at constant pressure cp to the specific heat at constant volume cv is k = cp/cv = 1.3. Also, the ratio of the differential maximum pressure reading hw, in in of water, to the maximum pressure in the pipe, in psia, equals 120/246.7 = 0.454. Using the expansion-factor curve in the ASME Fluid Meters, Y = 0.994 for ft = 0.5999, and the pressure ratio = 0.454. And, from the same reference, the meter area factor Fa = 1.0084 for a steel meter operating at 640°F. 

The Expansion Factor Y. This corrects for density changes between taps. Y depends on the adiabatic exponent (k = CJC ), absolute pressure ratio (AP/P ), and the ratio, p, between restriction diameter and the inside pipe diameter. For liquid flow, Y = 1.0. For gas flow, Buckingham derived the equation [3] ... 

Divide the specific volume of the inlet gas, v, by the square of the expansion factor Y. This gives an effective incompressible specific volume, while the liquid specific volume, v, is unaffected. 

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