ThermodynamicsのA-Zガイド、熱&アンプ、質量移動、流体工学

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Void Fraction is a very important parameter in multiphase flows, and in particular in two-phase gas-liquid flows. A very wide variety of techniques have been developed for such measurements, and extensive reviews are given by Hewitt (1978 and 1982) and, more recently, Leblond and Stepowski (1995). Here, we will briefly review the three most commonly used methods (namely quick-closing valves, gamma absorption and impedance) for average void fraction and the use of local probes for local void fraction measurements.

Quick-closing valve technique. In this method, valves are placed at each end of the section in which the void fraction is to be measured, the flow is set up to the required condition and the valves quickly closed to isolate the section, after which the liquid phase can be drained, and its volume determined. If the volume of the section containing the trapped fluid is known, then the void fraction can be determined. The valves can either be hand-operated (usually with a linking lever to ensure simultaneous action) or can be operated using solenoids. It is obviously preferable if the valves are of the ball type, ensuring continuity of bore in the pipe before the closure action is taken.

Though the measured void fraction is surprisingly insensitive to closure time, there must be an inherent inaccuracy in the method, due to changes of flow pattern while the valves are closing. Of course, another major disadvantage of this method is that the system has to be closed down for each measurement, and then restarted again afterwards.

Gamma ray absorption technique. A beam of gamma rays is attenuated by absorption and scattering according to the exponential absorption equation as follows:

(1) where I is the received intensity, I0 is the incident intensity, μ is the linear absorption coefficient and z is the distance traveled through the absorbing medium. To measure void fraction, the usual procedure is to determine the received intensities IL and IG when the beam passes through the channel full of the liquid and gas phases, respectively. The void fraction is then usually related to the intensity for the flowing situation (I) as follows:

(2) Of course, the void fraction determined by application of Eq. (2) is the average value over the path length. This is not necessarily the same as the cross-sectional average. This problem can be eased by using a beam traversing system, followed by integrating the void fraction over the cross-section, or by using multiple beams; a typical three-beam gamma densitometer is shown in Figure 1. Figure 1. Three-beam gamma densitometer system developed at 1NEL. [Hewitt (1982)].

Although widely applied, the gamma densitometer has a number of serious limitations. Care has to be taken to ensure that radiation safety precautions are properly observed. There is a fundamental statistical limitation in the gamma absorption method: if the count rate is R counts per second, then the standard deviation sR in the count rate, as a fraction of R, is given by:

(3) where τ is the counting time. Obviously, the fractional standard deviation goes down with increasing count rate and increasing count time, but can never be less than is given by Eq. (3). Thus, there are often compromises to be made in terms of counting time, and in the size of the sources used, which clearly govern the value of R. Another major source of error is associated with flow patterns in the channel. Thus, Eq. (2) only applies if the phases are homogeneously mixed, or if they exist in successive layers perpendicular to the beam. If the liquid and vapor exist in layers parallel to the beam, then the void fraction is given by:

(4) Obviously, this case is somewhat unlikely, but in most real systems there is a definite effect of void orientation. An extreme case is that of Slug Flow where, on a time-average basis, the void is fluctuating between two extremes corresponding to the slug and bubble regions. This fundamental problem can be to some extent overcome by applying Eq. (2) over short time intervals, and then to calculate an average void fraction from the succession of values generated. The intervals should clearly be sort compared to the fluctuation frequency within the system, and this sometimes gives problems in statistical accuracy, as determined by Eq. (3). This is a fundamental problem with the gamma absorption method which has to be realized up front and faced up to!

Impedance method. The principle of the impedance method is to measure the capacitance or conductance (or both) of the two-phase mixture, determined between electrodes which are placed around or within the flow. A whole variety of electrode systems have been proposed, ranging from plates mounted within the flow to successive ring electrodes mounted flush with the wall and multiple pairs of electrodes on the tube wall, between which the capacitance is measured in succession [see Hewitt (1978, 1982) for review and details]. The relationship between εG and admittance (the reciprocal of impedance) A is often calculated from the Maxwell (1881) equations: for a homogeneous dispersion of gas bubbles in the liquid, we have

(5) where Ac is the admittance of the gauge system when immersed in the liquid phase alone, and CG and CL are the gas and liquid conductivities if the conductivity is dominating, and the dielectric constants of the gas and liquid if the capacity is dominating. For liquid droplets dispersed in a gas, the Maxwell equations give:

(6) Equations can also be derived for other flow configurations (slug flow and annular flow). A comparison of the variation of admittance with void fraction for the various regimes is shown in Figure 2 It will be seen that there is a strong flow dependency on the relationship between void fraction and admittance, and this is the major problem with the capacitance technique. Nevertheless, it must be stated that for many systems, measurement of capacitance is a convenient, safe and cheap approach. It gives very rapid time response, and there are many applications where it can be useful. Figure 2. Effect of flow pattern on admittance [calculated by Bouman et al. (1974)].

The sensitivity to flow pattern can be reduced by using modern signal processing techniques, coupled with multiple wall-mounted electrodes, to produce tomographic images of the flow cross-section. These can be suitably averaged to give the mean void fraction, but the most important output is information on time variation of the phase content (which is, of course, related to the flow pattern). Work of this type is described by Xie et al. (1989).

Local void fraction measurements. It is of considerable scientific interest to measure the local value of void fraction in, say, bubbly flow. A review of such measurements is given by Delhaye (1982). Techniques for local measurements include conductivity probes, optical probes and thermal anemometer. Examples of the first two types are shown in Figures 3 and 4. Figure 3. Conductance probe for local void fraction measurement. [Delhaye (1982)]. Figure 4. Optical fiber probe for local void fraction measurement. [Delhaye (1982)].

In the conductance probe method, conductance between the probe tip and the tube wall is established when the probe is in the liquid phase, but the conduction path is broken when the probe is in the gas phase. Similarly, with the optical probe, light is released from the tip of the probe when this is in the liquid phase, but is not released (and passes to the photo transistor) when the tip of the probe is in the gas phase. With thermal anemometers, it is possible to make simultaneous measurements of local temperatures and void fraction [see Delhaye (1982)].

#### REFERENCES

Bouman, H., van Koppen, C. W. J. and Raas, L. J. (1974) Some investigations of the influence of the heat flux on the flow patterns in vertical boiler tubes, Paper A2, presented at the European Two-Phase Flow Group Meeting, Harwell, England, June 1974.

Delhaye, J. M. (1982) Local measurement techniques for statistical analysis of Handbook of Multiphase Systems, Ch. 10.3 G. Hetsroni, Ed., McGraw-Hill, New York.

Hewitt, G. F. (1978) Measurement of Two-Phase Plow Parameters, Academic Press, London. ISBN: 0-12-346260-6.

Hewitt, G. F. (1982) Measurement of void fraction. Handbook of Multiphase Systems, Ch. 10.2.1.2, G. Hetsroni, Ed., McGraw-Hill, New York. ISBN: 0-07-028460-1.

Leblond, J. and Stepowski, D. (1995) Some non-intrusive methods for diagnostics in two-phase flows, Multiphase Science and Technology, Vol 8, Begell House, New York.

Maxwell, J. (1881) A Treatise on Electricity and Magnetism, Clarendon Press, Oxford.

Xei, C. G., Plaskowski, A. and Beck, M. S. (1989) 8-Electrode capacitance system for two-component flow identification, Part 1: Thermographic imaging, Part 2: Flow regime identification, IEE Proc, 136 (A) No. 4, 173-190.

#### References

1. Bouman, H., van Koppen, C. W. J. and Raas, L. J. (1974) Some investigations of the influence of the heat flux on the flow patterns in vertical boiler tubes, Paper A2, presented at the European Two-Phase Flow Group Meeting, Harwell, England, June 1974.
2. Delhaye, J. M. (1982) Local measurement techniques for statistical analysis of Handbook of Multiphase Systems, Ch. 10.3 G. Hetsroni, Ed., McGraw-Hill, New York.
3. Hewitt, G. F. (1978) Measurement of Two-Phase Plow Parameters, Academic Press, London. ISBN: 0-12-346260-6.
4. Hewitt, G. F. (1982) Measurement of void fraction. Handbook of Multiphase Systems, Ch. 10.2.1.2, G. Hetsroni, Ed., McGraw-Hill, New York. ISBN: 0-07-028460-1.
5. Leblond, J. and Stepowski, D. (1995) Some non-intrusive methods for diagnostics in two-phase flows, Multiphase Science and Technology, Vol 8, Begell House, New York.
6. Maxwell, J. (1881) A Treatise on Electricity and Magnetism, Clarendon Press, Oxford.
7. Xei, C. G., Plaskowski, A. and Beck, M. S. (1989) 8-Electrode capacitance system for two-component flow identification, Part 1: Thermographic imaging, Part 2: Flow regime identification, IEE Proc, 136 (A) No. 4, 173-190.