The folowing note, by Tom Fortescue, describes two of the main sources of error when measuring multi-phase fluid density with a radiometric densitometer.

The two commonest problems are time distribution errors and spatial distribution errors.

Radiometric density measurements have to be averaged in order to reduce the measurement uncertainty. The resulting mean radiation absorption is then used to calculate the density, using a non-linear (exponential) calibration curve.

An important consequence of the non-linearity is that the density corresponding to the average of the radiation absorption is not the same as the average density. For the case of dispersed slug or large bubble flow, with 50% average void fraction, the indicated density can be 20% or more low, depending on pipe diameter.

This problem can be overcome by measuring so quickly that the transit time of the bubbles or slugs is significantly longer than the measurement integration time. Under these conditions, the fluid density is fairly constant over the duration of a single measurement, and an instantaneous density can be calculated from the individual measurements with reasonable accuracy. The average of the instantaneous densities will then be very close to the true mean density.

At a fluid velocity of 10m/s, a 100 mm long bubble will pass through the measurement plane in only 10 msecs. For accuracy under these conditions, the measurement integration time should not exceed about 2 msec.

On simple radiometric densitometers, the integration time typically lies between one and one hundred seconds. In consequence, relatively large errors in the indicated density must be expected from these instruments when the phases are dispersed, and the flow contains bubbles of gas, or slugs of liquid which are comparable in size to the pipe diameter.

The largest spatial distribution errors are caused by using a single, narrow measurement beam, which does not probe the whole cross-section of the measurement pipe. The sketches below illustrate four possible multiphase flow situations in a vertical pipe.

In the stratified flow example A, approximately 30% of the measurement beam length is in the gas phase, so the instrument will indicate a density of about 70% of the liquid density. However, due to the shape of the pipe, less than 20% of the section is filled with gas, so the true density will be nearer 80% of the liquid density. The indicated density would only be correct for a beam passing normally through a square section pipe.

In example B, the measurement beam completely fails to detect the gas phase, so the instrument will indicate 100% liquid density, although the gas phase may occupy nearly 30% of the pipe cross section. Similarly, in example C, the liquid phase is undetected, so the instrument will indicate gas phase density, when the true mean density could be as much as 30% of the liquid phase density.

Example D illustrates an annular liquid flow with a gas core occupying half the pipe diameter. The densitometer beam will indicate a density of 50%. However, if the core is approximately circular, it will actually occupy only 25% of the pipe cross section, so that the indicated density will be 33% below the true density.

For a liquid phase of water, or oil, these are errors of up to 250 kg/m³ or more, for a liquid phase density of 850-1000 kg/m³.

The above sketches show what might at first sight be regarded as 'pathological' examples. However, particularly in vertical pipes, with elongated bubble flows, very similar conditions can be observed in air-water loops with pipe diameters as small as one inch. They are more probable in pipes with larger diameters.

The spatial distribution problem could in principle be avoided by using a broad beam system, where the beam probed the whole of the pipe cross-section uniformly.

However, although the source may radiate uniformly in all directions, uniform probing is difficult to achieve due to pipe wall absorption. The part of the beam which passes through the axis of the pipe intersects the wall nearly normally, so has a relatively short path in the wall, and in consequence, a relatively low absorption by the wall.

However, the parts of the beam probing the volumes near the walls of the pipe pass through these walls at a fairly flat angle, so have a much longer path within the pipe walls, and a much higher absorption by the walls.

The plot on the right shows the resulting variation in sensitivity against distance of ray path from pipe axis, for a 100mm pipe with 10 mm thick steel walls, using a Cs137 source.

For this case, the contribution of the edges of the beam to the total measured intensity is too small to be significant. Density differences in the volumes probed by these parts of the beam will be under weighted, resulting in the possibility of spatial distribution errors which can still be relatively large, although they are typically at least a factor of two smaller than those for the narrow beam case.

Multi-beam densitometers offer two main advantages.

The weighting of the measurements of the individual beams can be chosen to provide partial compensation for the sensitivity variation over the pipe cross section.

However, much more importantly, any statistically significant differences between the densities indicated by the various beams, the 'chordal average densities', will provide information on the actual phase distribution. In conjunction with appropriate models, this information can be used both to reduce the error in the calculated mean density, and to increase the confidence in this value.

Thus, if a three beam instrument is used, in which the three beams between them probe most of the pipe cross section, and all three beams indicate the same density, then there is an extremely high probability that the flow is homogeneous.

If two beams indicate much the same density, but one of the peripheral beams indicates a significantly lower density, then a tilted stratified flow (case B above) must exist.

If the central beam indicates a lower density than the peripheral beams, this may indicate an annular distribution (case D), but if the measured densities are high, particularly in a non-vertical pipe, it is more likely that there are gas bubbles at the top of the pipe which are only being seen by the diametral beam.

Correct identification of the phase distribution is thus a function not only of the ratios between the densities indicated by the individual beams, but also of the absolute densities indicated.

Fairly sophisticated models are required to discriminate satisfactorily the phase distribution effects in the presence of the statistical fluctuations inherent in radiometric measurements.

In dispersed two-phase and multi-phase flows, density measurement errors in excess of 100 kg/m³ can be caused by both time and spatial phase distributions.

In order to minimize time distribution errors, at flow rates of around 10m/sec the measurement integration time needs to be 2 ms or less. At higher flow rates, it needs to be even shorter.

Spatial phase distributions errors are greatest when measurements are made with a single gamma or X ray beam which is narrow compared to the pipe diameter, so probes only a small part of the pipe cross section.

Some improvement can be realised using a broad measurement beam. However, this is still only a single measurement so can provide no information on the actual phase distribution.

Multi-beam instruments provide information on the actual phase distribution. This information can be used with phase distribution models both to reduce the absolute errors, and to reduce the measurement uncertainty bands.

T.R. Fortescue, 1997