Annular Two-phase Flow Research Articles

Predication of wall shear stress of vertical upward co-current adiabatic air-water annular flow in pipes

Annular flow has been widely applied to industrials such as oil extraction, refrigerators, nuclear reactors and so on. Wall shear stress is one of the most important parameters for studying two-phase annular flow. When calculating wall shear stress from interfacial shear stress, the relationship between the wall shear stress and the interfacial shear stress is complicated and needs some parameters such as film thickness which requires high-precision measurement technique to acquire accurately. Besides, some wall shear stress correlations developed for annular flow are applicable to flow conditions at atmospheric pressure only. Because of these reasons, four databases for vertical upward co-current adiabatic air-water two-phase annular flow have been collected with wide range of flow conditions and different pressure conditions. Using these data, a simplified model between wall shear stress and interfacial shear stress is proposed. The new model is derived from the force balance analysis of the liquid film and gas core. As void fraction of annular flow is close to 1, the relationship between wall shear stress and interfacial shear stress can be simplified to be linearly. Based on this model and existing interfacial shear stress correlations, the wall shear stress can be calculated directly from flow conditions and fluid properties that can be readily obtained from experiments. Also, a new wall shear stress correlation calculated from wave velocities has been suggested. It is developed from frictional pressure drop correlation. Correlations for two-phase multiplier and wave velocity as well as pure empirical correlations also have been compared with existing databases. From data comparison, the newly proposed correlations in this study agree well with experimental data and other correlations with good prediction accuracy have been pointed out.

Liquid-film thickness and disturbance-wave characterization in a vertical, upward, two-phase annular flow of saturated R245fa inside a rectangular channel

The liquid-film flow in a vertical, upward, two-phase annular flow of saturated R245fa is characterized experimentally under adiabatic conditions. The experiments are conducted inside a rectangular channel with a cross section of 11.6 × 36 mm2 for mass velocities ranging from 95 to 130 kg/m2s, vapor qualities from 0.63 to 0.9 and saturation temperature of 23 °C. Liquid-film thickness and disturbance-wave velocity and frequency are measured optically. Liquid-film thickness is recorded at a sampling frequency of 2000 Hz, while disturbance-wave velocity is recorded at a sampling frequency of 20 Hz. A parametric study as a function of quality and mass velocity is performed. Results show that the liquid-film thickness decreases linearly with increasing vapor quality. The liquid films investigated are very thin, and thinner than the critical liquid-film thickness below which momentum and mass transport are no longer driven by disturbance waves. Indeed, at high vapor quality, when the liquid film is very thin, the liquid-vapor interface becomes smoother and disturbance waves slow down and vanish. The frictional pressure gradient increases with quality until it reaches a peak. Results suggest that the subsequent decrease in pressure gradient is closely linked to the disappearance of disturbance waves. For the range of studied operating conditions, the frequency of disturbance waves remains constant within the uncertainty of the measurements. Experimental data is compared with prediction methods from the literature. Although reasonable predictions were obtained for the liquid film thickness, none of the tested methods could predict disturbance-wave velocity and frequency accurately.

Suppression of nucleate boiling in upward two-phase annular flow: A direct measurement using modern diagnostics

In heat exchangers employing flow boiling, e.g., boiling water reactors and industrial boilers, the heat transfer mechanism will transition from nucleate boiling to forced convective evaporation as the flow pattern transition from bubbly to annular flow at relatively high steam quality. Such thermal transition is called suppression of nucleate boiling (SNB). The occurrence of SNB affects the local heat transfer coefficient, the stability of liquid film, as well as the characteristics of entrained liquid droplets in the gas core. Despite its importance, there has been hitherto few direct measurements of the SNB conditions. Furthermore, the existing prediction approaches of SNB are only approximate, since they are based on extrapolation of empirical heat transfer correlations valid for nucleate boiling and forced convective evaporation regimes, rather than SNB mechanism. The objective of this study is to, experimentally and theoretically, fill the gap in understanding SNB phenomenon, using a modern set of diagnostics and a semi-empirical modeling approach. We leveraged synchronized infrared thermometry and an electrical conductance-based liquid film thickness sensor to investigate the details of the SNB phenomenon with high spatial and temporal resolutions. Such advanced diagnostics measure two crucial boundary conditions for SNB, i.e., the distribution of the temperature and heat flux on the heated wall, as well as the thickness of the liquid film. Such direct measurement revealed a clear dependency of SNB heat flux and wall superheat on both steam quality and mass flux. The experimental database has informed a more accurate semi-empirical model for predicting the SNB condition.

Experimental and numerical study of formation and development of disturbance waves in annular gas-liquid flow

Abstract Disturbance waves in a downwards annular gas-liquid flow were investigated experimentally and numerically in this study. In the experiment, the brightness-based laser-induced fluorescence (BBLIF) technique was utilized to obtain high-resolution spatiotemporal measurements for the film thickness. In the simulations, the two-phase system was simulated by the volume of fluid (VOF) method together with newly developed turbulence damping models, without which the turbulence level around the film surface is considerably under-predicted. Qualitative and quantitative comparisons were carried out for the experimental and numerical data, during which a novel method was developed to extract complex wave structures in a direct manner. Comparisons showed that the model is able to reproduce the main stages of flow evolution, including development of high-frequency initial waves, their coalesce into stable large-scale disturbance waves, generation of slow and fast ripples, and disruption of fast ripples into droplets. The main properties of modeled waves are in decent agreement with the measured ones, apart from noticeably rarer generation of ripples. The presented methods offer a new and promising option to model various energy technology systems, where annular two-phase flow occurs.

Wave structure and velocity in vertical upward annular two-phase flow

The knowledge of wave structure and velocity is of great importance to the prediction of annular two-phase flow phenomena. Given that most existing experimental studies focus on the low liquid flow rate, the goal of this study is to extend the understanding of wave characteristics to high liquid flow rate annular flow, i.e. wispy annular flow. Conductance film thickness probes have been used to measure the film structure of upward annular flow at a high temporal resolution. The inlet superficial gas velocity ranges from 10 m/s to 30 m/s and the range of inlet superficial liquid velocity is from 0.06 m/s to 2 m/s. Important parameters including average film thickness, wave height, base film thickness, wave frequency, wave velocity, and interfacial shear stress are measured and analyzed. Existing models for predicting these parameters are comprehensively reviewed and evaluated using the new data. The evaluation results indicate that although these models are developed for annular flow with low liquid flow rates, some of them can also be used in wispy annular flow with reasonable prediction accuracy, except for wave velocity models. The measured wave velocity in wispy annular flow is significantly higher than that in typical annular flow. A new correlation for wispy annular flow wave velocity is developed based on the new data.

Experimental characterization of boiling two-phase flow structures under BWR core operating conditions

Local two-phase flow parameters have been measured at the Westinghouse FRIGG facility in a test fuel bundle under Boiling Water Reactor (BWR) core operating conditions, over a wide range of flow and power (up to critical power). The measurements were performed at end of heated length using sapphire optical probes and Pitot tubes placed in various radial positions of the bundle. By using both instruments simultaneously, the void fraction, mixture velocity, dispersed phase (bubble or drop) characteristic lengths and volumetric interfacial area can be measured. In addition, the identification of two-phase flow regimes can be attempted. The measurements are of particular interest since there is very limited available data in the literature at high pressure relevant to BWR operation.Experimental measurements are reported in this paper covering a range of inlet flow and bundle power relevant to BWR operation, at a system pressure of 70 bar. The dynamic void fluctuations are used to establish two-phase flow regime maps, one based on global parameters (for 1-D applications) and the second based on local parameters (for CFD applications). However, it is discovered that transition flow regimes typically observed at low pressure (cap, slug, etc) are not similarly observed at high pressure. In addition, the measured characteristic lengths of the dispersed phase are reported in bubbly and annular two-phase flow regimes. In view of the results, recommendations are provided for further experimental work and modeling improvements.

Experimental study on radial evolution of droplets in vertical gas-liquid two-phase annular flow

In vertical gas-liquid two-phase annular flow, the radial evolution of droplets in the gas core plays a key role in affecting the characteristic of gas carrying liquids. In this paper, the droplets size and velocity at 11 radial positions were measured with the phase Doppler anemometry and then analyzed based on the recognition method of droplet role to figure out the radial evolution characteristics of droplets flow rate, size, velocity, and momentum. The results show that the droplet quantity flow rate decreases with the entrained droplets moving from the liquid film-gas core interface to the centerline, due to the fact that the droplet coalescence has a greater effect than the droplet breakup. The proportion of small droplets decreases while the proportion of relatively large droplets increases. The overall distribution characteristics of droplet diameter remain unchanged. The overall distribution characteristics of droplet axial velocity, radial velocity, and velocity size remain unchanged, although the droplets undergo the process of axial acceleration and radial deceleration during the radial migration. The distribution of droplet momentum varies greatly in the process of droplet radial migration. The contribution of small droplets to the total droplet momentum gradually decreases, while the contribution of large droplets to the total droplet momentum gradually increases. Although the number of large droplets is small, their contribution to the total droplet momentum cannot be neglected.

Experimental Investigation of Vertical Downward Two-Phase Flow in Annulus

Abstract A limited amount of work exists on gas–liquid flow in vertical pipe annulus, and, to the knowledge of the authors, there is no work on the literature to characterize vertical downward two-phase flow in pipe annulus. In the petroleum industry, downward two-phase in annulus is encountered on liquid-assisted gas-lift (LAGL) unloading and production operations. This study presents experimental data for pressure gradient, liquid holdup, and flow regimes for vertical downward two-phase (air and water) flow in pipe annulus. Also, the applicability of two-phase flow models are evaluated. The experimental results show that the liquid holdup is consistently higher for downward flow in annulus than in pipes for the annular flow regime, and these differences are as high as 45%. When the flow regime map for downward flow in annulus is compared with the ones in the literature for flow in pipes, it is observed that the intermittent flow in pipes occurs at lower liquid velocities than flow in annulus. The comparison between experimental data and model results also shows some discrepancy for liquid holdup and pressure gradient. These differences are high for annular and intermittent flow regimes, with errors of 100% for the liquid holdup and 200% for pressure gradient. However, the errors for bubble flow regime are much smaller, generally lower than 20%.

Effect of Viscosity on Pressure Drop of Oil-Water Two Phase Flow in 6” Horizontal and Inclined Stainless Steel Annulus Pipe

Two phase annular fluid flow (oil in water) commonly found in petroleum industry and is generally used for the lubricated transportation of highly viscous oil. Despite their importance, behavior of such flows has not been explored to an appreciable extent. Liquid viscosity may change considerably because of changes in temperature as in deep-water oil production. The variations in liquid viscosity have a remarkable influence on flow characteristics. In this experimental work, attention has been focused on effect of viscosity on pressure drop measurements of oil-water annular two-phase flow in a horizontal and inclined 6-inch diameter stainless steel pipe at different flow conditions. Two different mineral oils Exxsol D80 and Exxsol D130 along with water were used as working fluids in the present experimental study. Experiments were carried out for different inclination angles including: 0° - 90° and for different water cut (WC) ratios (0 to 100% in steps of 20 %). Inlet oil-water flow rates were varied from 2000 to 12000 barrels-per-day (BPD). For a given flow rate the frictional pressure drops (FPD) has been found to decrease from WC = 0 to WC 20%. Further increase in WC, FPD has been found to increase up to WC 40%. For WC more than 40%, the decrease in FPD is not appreciable. This could be due to phase inversion or change in flow pattern regime. For a given case with WC 40% (horizontal, ? = 0° case, flow rate = 10000 BPD), the FPD of Oil (D130) is more than FPD of Oil (D80) by 7%. This implies Oil (D80) is more suitable for transportation of oil-water two phase flows. For a given case (with WC 40%, 10000 BPD), for D80 oil, for increase in angle from zero degree to 90 degree, the increase in FPD is not appreciable. However, for the above scenario, for D130 oil, for increase in angle from zero degree to 90 degree, the percentage increase in FPD has been found to be 27%.

Characterization of Oscillations during Flow Boiling of Water in Parallel Microchannels

Several challenges coexist in the field of flow boiling in microchannels, ranging from high superheat required for boiling incipience to boiling instabilities and early dryouts. The aim of this study is to mitigate or solve some of the challenges and develop an image-processing algorithm for analysis of boiling oscillations in multiple parallel channels. The experimental results were acquired on an array of 64 parallel $$25\times 25$$ $$\mu$$ m microchannels using a synchronized high-speed visualization and measuring system. The small cross section of the microchannels allowed only the formation of annular two-phase flow, and a computer algorithm was developed for tracking the meniscus oscillations during boiling. The applied image analysis focuses on reliability with the simultaneous use of brightness variation and brightness derivative along with image subtraction. Moreover, the images were preprocessed to determine the number of microchannels and their orientation with applying different filtering and Radon transformations. The data extracted from the visualization helped determine the peak-to-peak amplitudes and fundamental frequencies of the oscillating meniscus. The results exhibit lower amplitudes and higher fundamental frequencies with increasing heat flux. The mass flux was kept constant at 83 kg/m $$^{2}$$ s, whereas the heat flux varied from 150 kW/m $$^{2}$$ to 250 kW/m $$^{2}$$ . The amplitudes and the fundamental frequencies of the meniscus oscillations determine the length and duration of microchannel with periodically alternating liquid and vapor phases.

Application of the image analysis on the investigation of disturbance waves in vertical upward annular two-phase flow

Annular two-phase flow commonly exists in industries process, and the disturbance wave at the gas-liquid interface plays an essential role in the mass, momentum and energy exchange. Although extensive experimental and analytical approaches to properties of disturbance waves have been implemented, most of the empirical correlations constructed to predict the wave properties have defects in prediction, especially under the high-pressure conditions. In the present paper, an image analysis method by acquiring the spatiotemporal distribution of the liquid film is employed to investigate the wave properties in a 20 mm I.D. tube in the upward annular regime. Because the interfacial waves should first penetrate through the gas boundary layer before full development, the effect of the gas boundary layer cannot be ignored and is undoubtedly enhanced when pressure increases. Accordingly, new empirical correlations for wave velocity and frequency are proposed by introducing the excess liquid Reynolds number. Compared with the experimental data from the literature, the proposed equations agree well with the experimental data within the averaged relative deviation of ±35%, especially under high-pressure conditions.

ON THE RELATION BETWEEN CHURN FLOW AND CRITICAL HEAT FLUX PREDICTION

Boiling crisis commonly occurs in many industrial systems, which is a threat to the security of the arrangements. Therefore, effective thermal management and accurate prediction of heat transfer are of utmost importance in the safety of heat transfer equipment. Although previous predictions of the critical heat flux (CHF) in heated annular two-phase flow had worked sufficiently well, the general assumption of the entrained fraction at the onset of annular flow is in contradiction to the reality, i.e., the discrepancies between the required and theoretical values become quite large, especially for higher mass flow rates. According to the discussions on the defects of the traditional liquid film flow model, the dryout mechanism relates more closely to churn flow. With great momentum, the entrained droplets in the churn flow travel along the pipe into the annular flow and inevitably affect the downstream flow field. Thus, it is more reasonable to start the calculation from the transition of the slug to the churn flow rather than the annular flow. The usual practice was to assume the entrained fraction at the onset of the churn flow. However, no published correlations have been successful in accounting for it. To address this deficiency, we discuss the transition of slug flow to churn flow and investigate the entrainment mechanisms and characteristics of droplets in the churn flow. Accordingly, we propose an empirical correlation to predict the entrained fraction at the onset of the churn flow. Using the newly developed model, the prediction of CHF in tubes yields good results. More importantly, the defects of the original model are solved with the help of reasonable physical interpretations, and the present model provides better understanding of the effect of flow pattern on CHF predictions.

Evaluación del espesor de una película de agua descendente vertical con visualización y técnicas de reconocimiento de imágenes

The present work is focused on the experimental study of a vertical downward annular flow to determine the thickness and stability of a falling film water flow. For this purpose, it was designed and implemented an experimental set up to generate the annular two-phase flow pattern, provided with an injection head with a special geometry to induce a liquid film inside a cylinder. Due to the small dimensions, the film was visualized using a pulsated laser to illuminate the region and the pictures were taken with a high-speed camera. This technique allowed the determination of the falling film thickness by means of an algorithm to recognize image contours. In some of the studied cases, a concurrent air flow was injected in the center of the cylinder in order to evaluate its influence on the interfacial hydrodynamics of the liquid film. Average film thickness were obtained for different Reynolds numbers in different axial observation points, and it was observed that the liquid film annular area and the shape of the header to inject the water, are important factors for the surface characteristic and thickness of the film, and its stability as well. The experimental results show that the standard deviation increases in proportion to the average film thickness, especially in the turbulent regime.

Characterization and development of an electrical conductivity flow cell for the axial continuous phase fraction profiling of bulk multiphase flow

In this study, the electric field generated by axial ring electrode conductivity cells is investigated. It is shown that the geometric parameters of such cells (mainly electrode separation, diameter and width) have a large effect on the resulting electric field uniformness. To quantify this effect, a parameter termed the effective measurement volume (EMV) is introduced which quantifies the measurement volume within the conductivity cell that contains 90% of the total current, normalized to that of an ideal conductivity cell (which generates a uniform electric field). The effects of a conductivity cells EMV on bulk conductivity measurements are both simulated and experimentally confirmed for emulated annular two-phase flow conditions. The results show that measurements performed with conductivity cells of high EMV tend towards true void fraction values, whereas cells of low EMV tended towards larger measurement error. This is quantified in the sum of squared residuals of the measurement, where a conductivity cell with an EMV of 0.19 has a sum of squared residual that is 638.4 times larger than a cell with an EMV of 1.00. Understanding the limitations of geometric parameters, a new axial conductivity profiling sensor was developed that uses one pair of electric field generating electrodes, and multiple measurement electrodes to create an axial conductivity profile with an axial measurement resolution of 10 mm.

A new entrainment rate model for annular two-phase flow

Reliable entrainment rate model is of great importance for the prediction of annular flow as it specifies the mass and momentum transfer between the liquid film and the droplet in the gas core. Various entrainment rate models have been proposed, yet few of them have a satisfying performance for a wide range of flow conditions, including gas/liquid phase flow rates, fluid type, channel size, etc. To develop a reliable entrainment rate model that is applicable to various fluids, instrumentation methods and available databases are reviewed in detail. The database for model development is determined by analyzing the instrumentation reliability and eliminating data points with the obvious error. The entrainment rate and the superficial liquid flow rate are nondimensionalized as Reynolds number using Laplace length as length scale, which is a critical wavelength of Taylor instability and related to the droplet size for annular flow. The entrainment rate model is developed based on the new nondimensional numbers, and the performance has been compared with various previous models. The evaluation results show that the prediction accuracy of the new model is better than previous models for a wide range of data.

Numerical analysis of pre-dryout sequences through the route of interfacial evolution in annular gas-liquid two-phase flow with phase change

Numerical modeling of two-phase annular flow boiling has been carried out inside a tube using volume of fluid based computational multifluid dynamics. The proposed numerical methodology has been validated with available experimental results from literature. The stages of bubble nucleation, growth, merging, bursting, droplet entrainment, film rewetting, and dryout has been clearly visualized for diabatic annular mist flow. Plots of liquid phase fraction and average heat transfer coefficient were plotted which shows decreasing trends with axial space. Effect of different degrees of superheats and gas-liquid velocities at a higher working pressure of 40 bar have been studied, which can be commonly observed in boiling water reactor (BWR) conditions. Dryout is found to happen early in the axial length if the degree of superheat and flow velocities increases. The chaotic interfacial behavior has been analyzed with mean, standard deviation and attractor plot of space time-varying values of liquid phase fraction.

Experimental investigation of sand particle erosion in a 90° elbow in annular two-phase flows

Upward gas-liquid-solid flows are often observed in oil and gas transmission pipelines. The presence of solids in such flows is a concern, as it can cause serious damage to pipe fittings, which are prone to erosion. These are normally components such as elbows and tees that can change the flow direction.The current work presents the results of an experimental erosion campaign including 56 tests in a 101.6-mm standard elbow (radius of curvature of 1.5) under gas-sand and gas-liquid-sand flow conditions. The latter included 38 tests (30 distinct conditions) mostly in the range of annular and unstable annular flows. Thickness loss measurements were made using a non-invasive ultrasonic technology instrument at 8 different points along the outer bend of the elbow to investigate effects of various factors on erosion. Additionally, flow visualization and paint erosion studies were performed to examine liquid distribution as well as the particulate phase behavior.Generally, when superficial liquid velocity increased, erosion ratio decreased for low liquid rate conditions. This trend, however, reversed for higher liquid rates. Also, the paint erosion experiments revealed that the location of the maximum erosion was around 40°–50° on the middle outer bend, for all cases examined.

Local liquid film behavior of annular two-phase flow on rod-bundle geometry – II. Modeling and verification

Local liquid film behavior of annular flow on rod bundle geometry is vital for accurate prediction of dryout. This paper is the Part II of a two-part study devoting to modelling and prediction of local liquid film behavior of annular flow on rod bundle geometry based on the unique feature of experimental phenomenon in Part I of this work. In this paper, a mechanism model is developed to predict the local liquid film thickness of annular two-phase flow on rod bundle geometry, in which the spatial distribution of gas flow in the cross-section of rod bundle is specified, and the local velocity of liquid film is obtained by employing the triangular relationship that taking into account properly the momentum and mass balances for the liquid film and the gas core. Ordinary differential equation (ODE) for the liquid film thickness is developed according to the Navier-Stokes equations for momentum balance of film flow on a rod to analyze the spatial characteristic of film thickness, in which the droplet entrainment-deposition process and the effect of circumferential shear stress are considered as boundary conditions during mass transfer at the gas-liquid interface. The present model predicts the experimental local liquid film thickness with the maximum relative error of ±10%. Circumferential coherent wave structure is formed by the combination effect of transversal stress gradient in the wave, droplet redeposition and film flow by circumferential shear stress, and the circumferential shear stress reduces the relative importance of the droplet redeposition with respect to the transversal stress gradient making the peripheral distribution of film thickness more uniform.

Local liquid film behavior of annular two-phase flow on rod-bundle geometry-I. Experimental phenomenon and analysis

Local liquid film behavior of annular flow on rod bundle geometry is vital for accurate prediction of dryout. This paper is the first part of a two-part study concerning measurement and analysis of local liquid film behavior of annular flow on rod bundle geometry. In this paper, air-water annular flow experiments have been performed at different gas and liquid superficial velocity conditions in a 3 × 3 simulating PWR fuel rod-bundle test-section, and a conductivity-based technique is developed to measure the local dynamical liquid film thickness. Large amplitude disturbance waves appear at all low liquid phase superficial velocity conditions through down-burst mechanism caused by the highly turbulent gas flow and strong inter-subchannel interaction. Wave shapes do not change drastically by the variation of superficial liquid velocity since liquid flow tends to fill out the rod bundle corner due to the nature of annular flow. For disturbance wave dominating the film flow case, secondary flow in subchannel produces a circumferential force to shear off the liquid film, and the liquid droplets are redistributed, resulting in the liquid film thickness for the central region being thicker than that of gap region. Cross-correlation analysis indicates that gap and central regions are in possession of circumferential coherent wave structure with the same interfacial velocity, and the wave position for the central region is a little ahead of that for gap region since a higher wave has a longer leeward and upwind faces. The current study on local liquid film behaviors for rod bundle and the possible mechanisms that causing the discrepancies in local liquid film structures can be further used for the mechanism model development for the liquid film flow on the rod bundle. Part II of this study will introduce a mechanism model as an alternative approach to local liquid film thickness prediction for annular flow in rod bundle geometry.

A study of two-phase annular flow using unsteady numerical computations

Flow behavior of thin liquid films in an annular flow regime is an important element of the thermal hydraulics of a BWR – it controls the heat transfer from the fuel rods to the coolant. Despite its relevance, the flow behavior of dynamic liquid films is not well understood even in adiabatic conditions. To investigate it, unsteady numerical simulations with an interface tracking model were performed in a double subchannel geometry with a P/D ratio of 1.325. Different turbulence models, namely the large eddy simulation (LES) and linear/nonlinear eddy viscosity unsteady-RANS (URANS) models, were tested. A novel approach for generating turbulent inlet conditions in a periodic flow domain was developed to reduce computational efforts. Validation against experimental data revealed shortcomings of the linear eddy-viscosity RANS model in predicting key flow parameters. By capturing the effects of the secondary flow structures in a subchannel geometry, improved predictions were obtained with a non-linear SST k–ω (QCR) turbulence model. Time-averaged liquid film thickness (LFT) and tracer distribution obtained with LES were found to have the best agreement with experimental data. Instantaneous and time-averaged velocity profiles were analyzed to understand the influence of the gas-liquid interface. Secondary flow structures in the subchannel gap region were found to enhance the turbulent mixing of the passive scalar in the liquid film. This finding is relevant towards the prediction of thermal-hydraulic parameters in a multichannel flow assembly by accounting for inter-channel mixing phenomena.