Measuring Phase Fraction with Tomography

One-sentence definition

Phase fraction is the volume fraction occupied by one phase in a multiphase flow — for example the “void fraction” in gas-liquid flow, the “water cut” in oil-water flow, or the “solids concentration” in gas-solid flow. It is the parameter that process control, flow modelling and safety monitoring care about most, and it has long been a measurement challenge.

Tomography (ECT / ERT in particular) measures the capacitance or conductance between electrodes and then, with the help of an EMA (Effective Medium Approximation) model, maps the electrical measurement to phase fractions. The whole process is non-invasive and real-time.

Why you can’t read it off directly

An electrical tomography sensor measures the capacitance or conductance between electrodes, which does not directly equal “30% void fraction” or “70% water cut”. Two layers of relationship sit between the measurement and the fraction:

• Geometry + medium → electrical quantity: the electrode structure and the spatial distribution of the medium together determine how much capacitance / conductance is measured between electrodes
• Electrical quantity → phase ratio: only once the phase-distribution pattern is known or assumed can the volume ratio of each phase be inferred

The second layer is exactly what the EMA (Effective Medium Approximation) model addresses: it relates “the effective electrical constant exhibited by a blob of mixed medium” to “the ratio of each phase + each phase’s intrinsic electrical constant”.

Five common EMA models

Let the intrinsic value of the high-constant phase be εh, the low-constant phase εl, the mixed effective value εe, and the high-constant-phase fraction αh. The common models are:

Model	Effective relation (two-phase case)	Applicability
Parallel	εe = αh·εh + (1−αh)·εl	Planar / coaxial laminar distribution; fraction is linear in capacitance
Serial	1/εe = αh/εh + (1−αh)/εl	Annular film flow; field runs along the series direction
Maxwell-Garnett (MG)	εe = εl · [2εl + εh − 2αh(εl − εh)] / [2εl + εh + αh(εl − εh)]	Low dispersed-phase fraction, isolated particles suspended in the continuous phase
Bruggeman	αh·(εh − εe)/(εh + 2εe) + (1−αh)·(εl − εe)/(εl + 2εe) = 0	Higher fraction, dispersed particles beginning to interact; typical gas-liquid
Böttcher	Assumes particles are surrounded by the mixed medium rather than the pure continuous phase; extends to high solids fraction	High solids fraction (dense suspension, fluidised bed)

The parallel and serial models can be regarded as the upper and lower bounds of all EMA models: for the same mixing state, the parallel model gives the highest estimated high-constant-phase fraction and the serial the lowest, with the other models (MG / Bruggeman / Böttcher) falling in between.

From image to fraction: two data-extraction paths

Tomography produces a cross-sectional image; there are two common ways to compute fraction from it.

Path A: reconstruct the image first, then apply an EMA model pixel by pixel

• Use LBP, Landweber, Tikhonov regularisation, etc. to reconstruct a normalised permittivity / conductivity distribution
• Apply the same EMA model to each pixel to convert the normalised value into a local phase fraction
• Finally integrate over the whole cross-section or the region of interest to obtain the total fraction

The advantage is that the whole cross-section uses one EMA model and the computation is simple; the prerequisite is that the reconstruction itself is accurate enough.

Path B: use the raw measurements directly, applying EMA models by group

• Skip full reconstruction and take the raw capacitance / conductance measurements between electrodes
• For each electrode pair (or each group of adjacent / opposite electrodes) pick an EMA model individually and map the measurement to a fraction
• Weight or combine the multiple results to obtain the average fraction

This approach is often more accurate when the flow regime is known (stratified, annular, mist…): different electrode-pair sensitivity regions correspond to different phase positions, so picking the right EMA avoids the soft-field error introduced by image reconstruction.

How to choose an EMA model

Pick the wrong model and the fraction error can be tens of percentage points off. Three things matter most:

• Flow regime / distribution pattern: stratified / annular → parallel, serial; dispersed particles / bubbles → MG (low fraction), Bruggeman (medium-high fraction), Böttcher (very high fraction)
• Two-phase electrical-constant ratio εh/εl: the larger the ratio, the more the models diverge and the more sensitive the result is to model choice; this ratio can serve as an uncertainty indicator
• Sensor structure + sensitivity distribution: single-pair, helical and ring electrodes each have typical applicable models; the multiple electrode pairs of ECT/ERT can even use different models by group

For industrial processes where the flow regime changes (e.g. gas-liquid transport going stratified → slug → mist), a single EMA model often cannot cover the whole operating range — you need to identify the flow regime online and switch EMA, or fuse multiple models by weighting.

Engineering practice notes

• Calibrate the sensor before use: take reference values for the single-phase full / empty pipe, normalise measurements to the 0-1 range before applying the EMA model — this cancels much of the electrode-geometry and parasitic effects
• Watch for electrical-constant drift with temperature / pressure: water’s permittivity changes markedly with temperature, so industrial sites need temperature compensation
• Avoid the εh ≈ εl regime: when the two-phase electrical constants are too close, fraction-inversion sensitivity drops sharply — supplement with another modality (ultrasound, microwave)
• Be careful with three-phase flow: a single ECT or ERT under-determines phase-fraction inversion in three-phase flow; use dual-modality ECT+ERT, or pair with ultrasound / Venturi

Want to go deeper?

The core conclusions of this article come from the review paper: Z. Cui, Q. Zhang, K. Gao, Z. Xia, H. Wang, “Electrical Impedance Sensors for Multi-Phase Flow Measurement: A Review”, IEEE Sensors Journal, Vol. 21, No. 24, Dec. 2021, pp. 27252–27267 (DOI: 10.1109/JSEN.2021.3124625). It gives the full derivation of the five EMA models, numerical comparison curves, and an application cross-reference across ECT, ERT, WMS, FFS and other sensor structures.

Next step

To build the overall framework first, read What Is Tomography; to understand how tomography infers fraction, read this article; for the selection stage see ERT vs ECT. If your process has an unusual flow regime or phase combination, contact us with your medium, flow regime and fraction range, and we will evaluate the EMA-model / sensor-structure combination together.