7.4 Electrical impedance tomography

Electrical impedance tomography (EIT) is a powerful tool for visualizing multiphase flows. By measuring electrical conductivity differences, EIT creates real-time images of phase distributions in various systems, from bubble columns to oil-water pipelines.

EIT's non-invasive nature and high temporal resolution make it ideal for monitoring dynamic processes. While spatial resolution is limited, advanced techniques like 3D imaging and dual-modality systems continue to push the boundaries of EIT's capabilities in multiphase flow analysis.

Electrical impedance tomography fundamentals

Principles of electrical impedance

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• Electrical impedance measures the opposition to alternating current (AC) flow in a material
• Impedance consists of resistance and reactance components
• Materials with different electrical properties exhibit varying impedance values
• Conductivity is the inverse of resistivity and quantifies a material's ability to conduct electric current
  • Conductivity is measured in siemens per meter (S/m)
  • Higher conductivity indicates lower impedance and easier current flow

Tomographic imaging techniques

• Tomography involves reconstructing cross-sectional images from projections or measurements taken at different angles
• Electrical Impedance Tomography (EIT) utilizes impedance measurements to create images of the internal conductivity distribution
• EIT injects small alternating currents through electrodes placed on the boundary of the object
• Resulting voltage measurements are used to estimate the internal conductivity distribution
• Mathematical algorithms reconstruct the conductivity image from the boundary measurements

Applications in multiphase flow

• EIT is well-suited for monitoring multiphase flows due to its sensitivity to conductivity differences between phases
• Multiphase flows involve the simultaneous presence of two or more phases (gas, liquid, solid)
• EIT can provide real-time visualization of phase distributions and flow patterns
• Applications include monitoring gas-liquid flows in bubble columns, oil-water pipelines, and fluidized beds
• EIT enables non-invasive measurement of phase fractions, bubble size distributions, and mixing characteristics

EIT system components

Current injection and voltage measurement

• EIT systems consist of a current injection unit and a voltage measurement unit
• The current injection unit applies small alternating currents through a set of electrodes
• Typical current amplitudes range from a few milliamps to tens of milliamps
• The voltage measurement unit records the resulting voltages across pairs of electrodes
• Voltage measurements are synchronized with the current injection pattern
• The number of independent voltage measurements depends on the number of electrodes and injection pattern

Electrode configurations and materials

• Electrodes are placed on the boundary of the object or vessel being imaged
• Common electrode configurations include adjacent, opposite, and cross patterns
• The number of electrodes determines the spatial resolution and sensitivity of the EIT system
  • Increasing the number of electrodes improves resolution but increases complexity
• Electrode materials should have low contact impedance and be chemically inert
• Stainless steel, silver, and silver/silver chloride are commonly used electrode materials

Data acquisition systems and hardware

• EIT data acquisition systems control the current injection and voltage measurement processes
• The system typically consists of a microcontroller or FPGA for timing and control
• Analog front-end circuitry includes current sources, voltage amplifiers, and multiplexers
• Analog-to-digital converters (ADCs) digitize the measured voltages for further processing
• The data acquisition system communicates with a computer for image reconstruction and display
• Specialized EIT hardware is available commercially or can be custom-built for specific applications

Image reconstruction algorithms

Forward and inverse problem formulation

• EIT image reconstruction involves solving an inverse problem to estimate the conductivity distribution from boundary measurements
• The forward problem calculates the expected voltage measurements given a known conductivity distribution
• The inverse problem estimates the conductivity distribution that best fits the measured voltages
• The forward problem is typically solved using finite element methods (FEM) to model the current flow and potential distribution
• The inverse problem is ill-posed and requires regularization techniques to obtain stable solutions

Sensitivity matrix and Jacobian calculations

• The sensitivity matrix, also known as the Jacobian matrix, relates changes in conductivity to changes in voltage measurements
• Each element of the sensitivity matrix represents the sensitivity of a voltage measurement to a change in conductivity at a specific location
• The sensitivity matrix is calculated by solving the forward problem for small perturbations in conductivity
• The Jacobian matrix is used in iterative image reconstruction algorithms to update the conductivity estimate
• Efficient computation of the Jacobian matrix is crucial for real-time image reconstruction

Regularization techniques for stability

• Regularization techniques are employed to stabilize the ill-posed inverse problem in EIT
• Tikhonov regularization is a common approach that adds a penalty term to the objective function
  • The penalty term promotes smoothness or other desired properties in the reconstructed image
• Total Variation (TV) regularization preserves sharp edges and boundaries in the image
• Laplacian regularization encourages spatially smooth conductivity distributions
• The regularization parameter controls the balance between data fitting and prior assumptions
• Proper selection of the regularization technique and parameter is important for obtaining accurate and stable reconstructions

Multiphase flow monitoring with EIT

Conductivity differences in multiphase mixtures

• EIT exploits the conductivity differences between phases to visualize multiphase flows
• In gas-liquid systems, the gas phase has a much lower conductivity than the liquid phase
• Liquid-liquid systems (oil-water) exhibit conductivity contrasts due to the different ionic content of the liquids
• Solid-liquid suspensions have conductivity variations depending on the particle concentration and properties
• The conductivity contrast between phases allows EIT to distinguish and quantify the phase distributions

Gas-liquid and liquid-liquid systems

• EIT is widely applied in gas-liquid systems such as bubble columns and airlift reactors
• The technique can measure gas hold-up, bubble size distribution, and flow regime transitions
• In oil-water pipelines, EIT monitors the phase fractions and flow patterns (stratified, dispersed, or slug flow)
• EIT provides real-time information for optimizing separator design and operation
• Liquid-liquid extraction processes benefit from EIT monitoring of phase dispersion and mixing efficiency

Solid-liquid suspensions and slurries

• EIT is used to characterize solid-liquid suspensions and slurries in various industrial processes
• The technique can monitor particle concentration, homogeneity, and settling behavior
• In hydrocyclones, EIT visualizes the separation of solid particles from the liquid phase
• Slurry pipeline transport can be optimized by monitoring the solids distribution and detecting blockages
• EIT enables non-invasive measurement of local solids concentration profiles in stirred tanks and crystallizers

Advantages and limitations of EIT

Non-invasive and non-intrusive nature

• EIT is a non-invasive imaging technique that does not require physical penetration into the system
• The electrodes are placed on the boundary of the vessel or pipe, minimizing flow disturbance
• Non-intrusive measurement allows continuous monitoring without interrupting the process
• EIT is suitable for opaque systems where optical techniques are not applicable
• The technique can be applied to hostile environments (high temperature, pressure, or corrosive media)

High temporal resolution vs spatial resolution

• EIT offers high temporal resolution, enabling real-time monitoring of dynamic processes
• Data acquisition rates can reach hundreds of frames per second, capturing fast transient phenomena
• However, the spatial resolution of EIT is limited compared to other tomographic techniques (X-ray, MRI)
• The spatial resolution depends on the number of electrodes and the sensitivity of the measurements
• Increasing the number of electrodes improves spatial resolution but increases system complexity and cost

Sensitivity to conductivity changes and distributions

• EIT is highly sensitive to changes in conductivity within the imaging domain
• The technique can detect small variations in conductivity caused by phase distributions or concentration gradients
• EIT is more sensitive to conductivity changes near the boundary (close to the electrodes)
• Sensitivity decreases towards the center of the imaging domain, affecting the reconstruction accuracy
• The sensitivity distribution can be improved by optimizing electrode placement and injection patterns

Advanced EIT techniques

3D and multi-plane imaging

• Conventional EIT provides 2D cross-sectional images of the conductivity distribution
• 3D EIT extends the imaging capabilities by using multiple electrode planes or 3D electrode arrays
• 3D reconstruction algorithms incorporate information from multiple planes to generate volumetric images
• Multi-plane EIT systems can capture the axial variation of conductivity along the flow direction
• 3D EIT enables visualization of complex flow structures and asymmetric distributions

Dual-modality systems with ECT or ultrasound

• Dual-modality systems combine EIT with other tomographic techniques for enhanced imaging capabilities
• Electrical Capacitance Tomography (ECT) is often integrated with EIT for imaging non-conductive phases
• ECT measures the permittivity distribution, complementing the conductivity information from EIT
• Ultrasound tomography can be combined with EIT to obtain additional velocity or density information
• Dual-modality systems provide a more comprehensive characterization of multiphase flows
• Data fusion techniques are employed to combine the information from different modalities

Adaptive and dynamic image reconstruction

• Adaptive EIT reconstruction algorithms adjust the imaging parameters based on the flow conditions
• Dynamic reconstruction techniques update the conductivity estimate as new measurements become available
• Kalman filtering and recursive least squares methods are used for real-time image reconstruction
• Adaptive mesh refinement strategies improve the resolution in regions of interest or high conductivity gradients
• Machine learning techniques, such as neural networks, can be applied to enhance image quality and interpretation
• Dynamic EIT imaging captures the temporal evolution of conductivity distributions in transient flows

Practical considerations for EIT implementation

Electrode placement and contact impedance

• Proper electrode placement is crucial for obtaining accurate and reliable EIT measurements
• Electrodes should be evenly distributed around the boundary of the imaging domain
• The distance between electrodes affects the sensitivity and resolution of the measurements
• Electrode size and shape influence the current density and measurement stability
• Contact impedance between the electrodes and the medium should be minimized
• Techniques such as electrode surface treatment or conductive gels can improve electrode-medium contact

Calibration and reference measurements

• EIT systems require calibration to account for system-specific parameters and electrode variations
• Reference measurements are taken with known conductivity distributions to establish a baseline
• Calibration phantoms with well-defined conductivity patterns are used to assess system performance
• Regular calibration helps maintain the accuracy and stability of EIT measurements over time
• Online calibration techniques can compensate for drift or changes in electrode properties during operation

Noise reduction and signal processing strategies

• EIT measurements are susceptible to various noise sources, including electrical interference and contact impedance fluctuations
• Proper shielding and grounding of the EIT system can reduce electromagnetic interference
• Signal processing techniques, such as filtering and averaging, are applied to improve the signal-to-noise ratio
• Adaptive noise cancellation methods can suppress specific noise components (e.g., power line interference)
• Optimal frequency selection for the excitation current minimizes the impact of capacitive and inductive effects
• Advanced signal processing algorithms, such as wavelet denoising or principal component analysis, can enhance image quality

Case studies and industrial applications

Bubble columns and fluidized beds

• EIT has been extensively applied to study gas-liquid flows in bubble columns and fluidized beds
• The technique provides insights into bubble size distribution, gas hold-up, and flow regime transitions
• EIT measurements help optimize gas distributor design and operating conditions for improved mass transfer
• Real-time monitoring of fluidized beds enables detection of channeling, slugging, or bed collapse
• EIT data can be used to validate and improve computational fluid dynamics (CFD) models of multiphase reactors

Oil-water pipeline flow monitoring

• EIT is employed in the oil and gas industry for monitoring multiphase flows in pipelines
• The technique can measure the phase fractions and identify flow patterns (stratified, dispersed, or slug flow)
• Real-time EIT monitoring allows for early detection of water breakthrough or excessive water production
• EIT data aids in optimizing pipeline design, flow control, and separation processes
• Integration of EIT with other sensors (pressure, temperature) provides a comprehensive flow characterization

Mixing and reaction vessels in chemical processing

• EIT is used to monitor mixing processes in chemical reactors and stirred tanks
• The technique can visualize the spatial distribution of reactants and products
• EIT measurements help assess mixing efficiency, dead zones, and short-circuiting
• Real-time monitoring of reaction progress enables optimal control of process parameters (temperature, agitation speed)
• EIT can detect the formation of precipitates or solid deposits in crystallization processes
• Integration of EIT with process analytical technology (PAT) tools enhances process understanding and control