Basic Flow Measurement Laws

The chapter goes through the laws which should be recognized and obeyed before flow measurement is attempted. Certain physical laws explain what happens in the “real” world. Some of these laws explain what happens when fluid flows in a pipeline, and these in turn explain what happens to a flowing stream as it goes through a meter. The laws discussed in this chapter include the law of “conservation of mass”, also called the “Law of Continuity,” the law of “conservation of energy”, the Fluid Friction Law and the fundamental flow equation. The Reynolds number is explained, as are the gas laws and ways to correct for the effects of temperature and pressure.

Keywords

physical law; fluid flow; conservation of mass; conservation of energy; fluid friction law; fundamental flow equation; Reynolds number

All of the following laws should be recognized and obeyed before flow measurement is attempted. Certain physical laws explain what happens in the “real” world. Some of these laws explain what happens when fluid flows in a pipeline, and these in turn explain what happens to a flowing stream as it goes through a meter. All variables in the equations must be in consistent units of measurement.

The law of “conservation of mass” states that the mass rate is constant. In other words, the amount of fluid moving through a meter is neither added to nor taken from as it progresses from point 1 to point 2 (Figure 2-1). This is also called the “Law of Continuity,” and it can be written mathematically as follows:

1=M2 (2.1)

where:

Since mass rate equals fluid density multiplied by pipe area multiplied by fluid velocity, Equation 2.1 can be rewritten as:

1A1V1=ρ2A2V2 (2.2)

where:

In terms of volume rate this can be restated as:

=AV (2.3)

where:

The law of “conservation of energy” states that all energy entering a system at point 1 is also in the system at point 2, even though one form of energy may be exchanged for another. (Note: the Bernoulli theorem relates the same physics in fluid mechanics.) The total energy in a system is made up of several types:

The Fluid Friction Law states that energy is required to overcome friction and move fluid from point 1 to point 2. For the purpose of calculating flows, certain assumptions are made about the stability of the system energy under steady flow. The main energy concerns are the potential and kinetic energies (definitions 1 and 3); the others are either of no importance, do not change between position 1 and position 2, do not occur, or are taken care of by calibration procedures. A generalized statement of this energy balance is given below:

E1+PE1=KE2+PE2 (2.4)

Kinetic energy (KE) is energy of motion (velocity). Potential energy (PE) is energy of position (pressure).

In simple terms, this equation can be rewritten:

1+VE1=PE2+VE2 (2.5)

where:

Equation 2.5 is the “ideal flow equation” for a restriction in a pipe. In real applications, however, certain corrections are necessary. The major equation correction is an efficiency factor called the “coefficient of discharge.” This factor takes into account the difference between the ideal and the real world. The ideal equation states that 100% of the flow will pass an orifice with a given differential, when in fact empirical tests indicate that a lower fraction of the flow actually passes for a given differential—for example, about 60% with differential between flange taps on an orifice meter, 95% across a nozzle, and 98% across a Venturi (Figure 2-2). This is caused by the device’s inefficiency or the loss from inefficiency caused by turbulence at the device where energy of pressure is not all converted to energy of motion. This factor has been determined by industry studies over the years and is reported as a “discharge coefficient.”

Equations 2.4 and 2.5 assume no energy, such as heat, is added or removed from the stream between upstream and the meter itself. This is normally of small concern unless there is significant difference between the flowing and ambient temperatures (i.e., steam measurement), or when measuring a fluid whose volume is sensitive to very small temperature changes, such as near its critical temperature. (Three common examples are ethylene, carbon dioxide gas, and hot water near its boiling point.)

It is also assumed that no temperature change is caused by fluid expansion (because of the lower pressure in the meter) from the upstream pressure to the meter. The small pressure difference between the two locations normally makes this theoretical consideration insignificant. If there is a change in state (i.e., from liquid to gas or gas to liquid) then this “insignificant” temperature change is no longer insignificant. Furthermore, the volume occupied (assuming no mass holdup) is much greater in the gaseous phase than the liquid phase; volume ratios of gas to liquids are as much as several hundred for some common fluids. Because of these problems, flow measurement of flashing liquids or condensing gases should not be attempted.

Reynolds Number

The Reynolds number is a useful tool for relating how a meter will react to a variation in fluids from gases to liquids. Since an impossible amount of research would be required to test every meter on every fluid we wish to measure, it is desirable that a relationship between fluid factors be known. Reynolds’ work in 1883 defines these relationships through his Reynolds number, which is defined by the equation:

=ρDvµ (2.6)

where:

Note: All parameters are given in the same units, so that when multiplied together they all cancel out, and the Reynolds number has no units. Units in the pound, foot, second system are shown below:

Based on Reynolds’ work, the flow profile (which affects all velocity-sensitive meters and some linear meters) has several important values. At values of 2,000 and below, the flow profile is bullet-shaped (parabolic). Between 2,000 and 4,000 the flow is in the transition region. At 4,000 and above the flow is in the turbulent flow area and the profiles are fairly flat. Thus, calculation of the Reynolds number will define the flow velocity pattern and approximate limits of the meter’s application. To completely define the meter’s application there must be no deformed profiles, such as after an elbow or where upstream piping has imparted swirl to the stream.

These effects will be further discussed in the sections covering the description and application of different meters, in Chapters 8, 9, and 10, and the equations will be covered more thoroughly later in this book.

These equations can be combined and rewritten in simplified forms. However, it is important to recognize the assumptions which have been made, so that if a metering situation deviates from what has been assumed, a “flag will go up” to indicate that the effect of Reynolds number must be evaluated and treated.

Gas Laws

Gas properties are almost always measured at conditions other than...