Robust computational framework for wire-mesh sensor potential field calculations

Flow Measurement and Instrumentation 67 (2019) 107–117

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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst

Robust computational framework for wire-mesh sensor potential field calculations

T

Corey E. Clifforda, Nolan E. MacDonalda, Horst-Michael Prasserb, Mark L. Kimberc,∗ a

Texas A&M University, Department of Nuclear Engineering, AIEN M104, 3133 TAMU, College Station, TX, 77843, USA ETH Zürich, Laboratory of Nuclear Energy Systems, ML K 13, Sonneggstrasse 3, 8092, Zürich, Switzerland c Texas A&M University, Department of Nuclear Engineering, Department of Mechanical Engineering, AIEN 205D, 3133 TAMU, College Station, TX, 77843, USA b

A R T I C LE I N FO

A B S T R A C T

Keywords: Bubbly flow Electric potential field OpenFOAM Two-phase flow Wire-mesh sensor (WMS)

With the development of the next generation of nuclear reactor safety system codes fast underway, increased importance has been placed on enhancing physical closure correlations and amassing representative benchmarkquality experimental data for validation purposes. Wire-mesh sensors, a reputable experimental measurement technique with sufficient spatial and temporal resolution to serve such goals, and related data reconstruction algorithms have been the subject of renewed interest as researchers attempt to characterize their measurement uncertainty. To assist in such investigations, the present work establishes a comprehensive numerical framework with which to quantify the electric potential field around wire-mesh sensors. Using the finite-volume foundations of OpenFOAM, a numerical solution algorithm is developed to predict the transmitted electric current between transmitter and receiver electrodes for both homogeneous and heterogeneous electrical conductivity fields. A detailed verification against seminal numerical calculations and robust validation procedure is included to ensure the accuracy of the proposed methodology. Parametric studies of spherical bubble diameter, lateral crossing position, and spheroidal shape influence are conducted to provide preliminary insights into wire-mesh sensor operation and the suitability of various calibration approaches. Observed trends in the transmitted currents reveal overshoots relative to calibration conditions, which are fundamentally linked to the maldistributed electric potential field in heterogeneous bubbly flows. The present investigation offers a vital first step towards a comprehensive multi-physics model of multiphase flow around a wire-mesh sensor.

1. Introduction In an effort to take advantage of recent developments in computer architecture, advanced numerical techniques, and enhanced physical models, the United States Department of Energy has commissioned the development of the next iteration of the Reactor Excursion and Leak Analysis Program (RELAP) system safety code, RELAP-7. Based on Idaho National Laboratory's Multi-Physics Object-Oriented Simulation Environment (MOOSE) framework, RELAP-7 expands upon the predictive abilities of previous code variants by employing a seven-equation, two-phase flow model capable of capturing the broad spectrum of phenomena occurring in modern light-water nuclear reactors. Despite these progressive modeling advancements, the averaging process utilized to homogenize the aforementioned balance equations prompts the loss of physical information and results in an underdefined set of governing expressions [1]. To fully constrain the governing equations and restore information lost during the ensemble averaging, semi- and even

∗

fully-empirical closure correlations need to be implemented in conjunction with the balance relationships. For the types of two-phase flows present in modern nuclear reactors, these closure correlations are required to describe the interaction with the model boundaries (wall drag and thermal transfer) as well as the interphasic heat and mass transfer. In general, each of these closure correlations are functions of the multiphase flow topology, which in turn are strongly dependent on the local phasic velocities, thermophysical properties, and geometrical configuration [2]. Although their inclusion is required for the outlined two-phase formulation, the addition of these empirical closure correlations diminishes the generality of the safety analysis code, thus heightening the need for a comprehensive validation procedure prior to wide-scale adoption. To aid in this validation endeavor, it is critical to develop benchmark quality experimental data sets in conditions akin to those experienced in light-water reactors with spatial and temporal resolutions sufficient to evaluate the performance of the applied closure correlations and assess the overall predictive capability of the nuclear

Corresponding author. E-mail address: [email protected] (M.L. Kimber).

https://doi.org/10.1016/j.flowmeasinst.2019.04.009 Received 27 November 2018; Received in revised form 20 February 2019; Accepted 15 April 2019 Available online 19 April 2019 0955-5986/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Wire-mesh sensor (WMS) schematic and electrode circuit diagram [6].

response of such sensors, especially in the presence of highly non-linear electrical conductivity fields (bubbly flows). To address such fundamental issues [7], constructed and evaluated finite-volume representations of wire-mesh sensors to characterize their response in multi-phase bubbly flow topologies. Within their models, the authors employed a uniform grid of cubical control volumes, with separate zones for receiver and transmitter electrodes, to determine the sensor's response to a discrete phase of spherical bubbles (both single and swarms) rising through a stagnant continuous medium. Bubbles were modeled by reducing the electrical conductivity in each of the affected control volumes. The conductivity of a given cell was calculated by counting the cells not covered by the bubble on a subgrid of 4 × 4 × 4 voxels and relating it to the number of subgrid cells, i.e., 64. Normalized dimensionless quantities were used for all of the simulations. Using said numerical approach, overshoots (relative to the calibration data sets) were observed in the computed electric current at the surface of the receiver electrodes, a phenomenon attributed to a distortion of the potential field symmetry as bubbles pass through the sensor. As a bubble of the discrete phase partially occupies a measurement voxel, the symmetric potential field between the planes of electrodes is fragmented, causing a rise in the local conductance and transmitted electric current relative to a completely continuous phase measurement. Despite the comprehensiveness of this research initiative, its findings are rooted by the principal assumptions of the investigation: (1) the application of linear stacks of cubic finite-volume cells to approximate cylindrical electrodes; (2) the boundaries of the computational domain (bottom and top of the square column) are receded sufficiently from the measurement positions as to not influence the computed currents; (3) the determination of the cell-averaged electrical conductivity via the previously detailed 4 × 4 × 4 subgrid method has a marginal effect on the predicted sensor behavior. Serving as a natural evolution to the work of [7]; the present investigation aims to assess the validity of each of these assumptions by developing a robust framework with which to evaluate electric potential fields around wire-mesh sensors. Using OpenFOAM's open-source, finite-volume computational fluid dynamics (CFD) toolbox, the presented computational solver takes full advantage of modern numerical conveniences (parallel domain decomposition, multigrid solution algorithms, universal mesh handling, etc.) in simulating wire-mesh sensors and their response to non-linear electric conductivity fields. To

systems code [3]. With an emphasis on establishing high-quality experimental data sets with characteristics suitable to serve as validation metrics for nuclear systems codes, recent research efforts have been focused on developing innovative measurement techniques and expanding the capability of proven methods. One such experimental measurement apparatus, the wire-mesh sensor (WMS), has a rich history in these types of investigations over the past three decades. First conceived by Ref. [4] to quantify the instantaneous concentration of water in a crude oil pipeline, wire-mesh sensors, and related data reconstruction algorithms, have since evolved to accommodate the requirements of benchmark experimental data sets. Modern wire-mesh sensor systems, as proposed by Ref. [5]; are principally based around two consecutive planes of electrodes (transmitters and receivers) oriented orthogonally to each other and offset by some marginal distance (see Fig. 1). During operation, each transmitter electrode is activated in sequential order with symmetrical, DC-free rectangular voltage pulses. Driver circuits supplying the transmitters are constructed with low output impedance relative to that of the electrode network immersed in the monitored fluid. Receiver electrodes are connected to the low impedance inputs of transimpedance amplifiers. Both considerations are necessary to suppress crosstalk caused by parasitic electrical currents which form between active and inactive electrodes. With the activation of each transmitter electrode (typically lasting 6–100 μs, depending on the temporal resolution of the employed data acquisition unit), an electric potential field forms between the energized transmitter and plane of receiver electrodes. The resulting electric current, the magnitude of which is proportional to the electrical conductivity of the surrounding medium, is collected consecutively at each receiver electrode. Although an electric current can take any viable path between the transmitter and receiver electrodes, the measured current signal is dominated by electrons traversing the shortest gap between the two wire planes. As a result, measurement voxels centered about the “crossing points” of the orthogonal electrode grid produce a two-dimensional matrix of electric current values at each discrete time level. Variations of the electrical conductivity within the measurement voxel, caused by a local change in the phasic composition, result in a shift of the measured current which can be identified after the application of a reconstruction algorithm. Prior to the application of wire-mesh data towards the development of enhanced physical models, it is imperative to understand the signal 108

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2.1. OpenFOAM solver Serving as the foundation of the present investigation, the aforementioned finite-volume solver is based in C++ and developed to work in conjunction with standard OpenFOAM utilities and file structure. Mimicking the operation of an actual wire-mesh sensor measurement sequence, the solution algorithm begins with a calibration cycle where nominal currents (continuous phase only) are computed for each transmitter-receiver intersection voxel. Subsequent to this preliminary calibration, the discrete phase is introduced to the simulation based upon a group of user-defined parameters linked via Equation (2):

(y − yb )2 (x − xb)2 (z − z b)2 + + =1 2 2 a b c2

where xb , yb , and z b define the centroid of the discrete phase while a , b , and c uniquely constrain the radius of the bubble in the x , y , and z Cartesian directions, respectively (see Fig. 2 for an illustrative summary of these parameters). Using this generalized definition, the current solver formulation can investigate spherical (a = b = c ), spheroidal (a = b ≠ c ), and generic ellipsoidal (a ≠ b ≠ c ) discrete phase bubbles. To translate the position of the discrete phase into the finite-volume computational domain, a two-stage Monte Carlo method is employed. First, each node of the finite-volume mesh is evaluated with Equation (2); if the node of a particular cell falls within the extent of the discrete phase, that cell is tagged with a unique identifier and processed further. Identified cells continue into a pseudorandom Monte Carlo procedure which determines the percentage of control volume occupied by the bubble. Once this parameter is estimated, the electrical conductivity of each cell is updated assuming linearity between the two phases. With a properly defined electrical conductivity field, both in terms of magnitude and spatial distribution, the remainder of the OpenFOAM solution algorithm follows the standard wire-mesh sensor measurement cycle. Using the zonal statistics present in the computational mesh (number of transmitter and receiver electrodes), the solution algorithm successively “activates” transmitter electrodes by manually specifying electric potential values at appropriate cell faces. Although any magnitude of electric potential can be employed within the present framework, the current study utilizes values of zero or unity to identify inactive and active transmitters, respectively. Using the finite-volume capabilities of OpenFOAM, Gauss's law (Equation (1)) is computed at each control volume based on the currently active transmitter electrode and discrete phase positioning. From the resolved electric potential field, the total current conducted to a receiver can be determined via the integral sum of the current density vector J (−σ ∇ϕ) over the surface of the electrode. Within the present solution algorithm, the transmitted electric current is approximated with Equation (3):

Fig. 2. Simplified problem geometry for insulating discrete phase bubble rising through stagnant conductive medium.

demonstrate the solver's efficacy, the described electric potential field algorithm is applied to a cardinal multiphase scenario: a discrete bubble traversing through a stagnant square column of a continuous medium (illustrated in Fig. 2 with germane geometric nomenclature). Using the results of [7] as a basis of comparison, an extensive verification and validation exercise is conducted to ensure the adequacy of the presented results and to quantify any associated numerical uncertainty. Once validated, parametric variations to several system constraints, namely discrete bubble size (diameter), shape (spherical or ellipsoidal), lateral crossing position, and electrode shape profile (cylindrical or rectangular), are considered to gauge the sensitivity of the system to each of these variables. The present study represents the critical first step towards the development of a comprehensive, fully-coupled modeling methodology with which to analyze wire-mesh sensor behavior within multiphase flow. 2. Computational methodology

I=−

∬ σ∇ϕ dA ≈ ∑ σ∇ϕ·Sf A

In order to quantify the passage of current between transmitter and receiver electrodes, the electric potential field must be resolved at all points within the computational domain. Employing the electroquasistatic (EQS) approximation in a charge-free environment (i.e., the electric field is irrotational and temporally-invariant during a single measurement cycle), the standard formulation of Gauss's law reduces to the second-order Laplacian expression present in Equation (1):

∇⋅(σ ∇ϕ) = 0

(2)

f

(3)

where Sf is the surface area vector of a cell face at the boundary of the given receiver. To evaluate the conducted current between the active transmitter and plane of receivers, Equation (3) is applied to each electrode sequentially which generates a scalar current for that particular “crossing point”. At that moment, the active transmitter potential is set to zero while the next in sequence is activated (ϕ = 1) and the procedure is repeated. Once the cycle for the final transmitter is completed, a two-dimensional matrix of electric currents is populated to characterize the wire-mesh sensor response for that particular temporal realization of the electric current. Finally, the axial position of the bubble (z b ) is updated based upon the user-defined time step size and bubble velocity and the entire procedure repeats until the discrete phase completely exits the computational domain. An illustration of the aforementioned solution algorithm, complete with visual representations of the various sequential components, is displayed in Fig. 3. For the sake of completeness, it should be explicitly noted that the presented finite-volume solution algorithm focuses solely on the electric

(1)

where σ and ϕ are the electrical conductivity and electric potential, respectively. As the discrete (bubble) and continuous (stagnant fluid) phases are considered to have unique values of electrical conductivity, the variable σ is treated as a field parameter and is calculated depending on the local composition within a control volume. To evaluate the electric potential around an arbitrarily sized, three-dimensional wire-mesh sensor, Equation (1) is discretized using the finite-volume technique and solved via a custom algorithm developed using OpenFOAM's open-source toolbox. 109

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handled within the finite-volume solution algorithm, the remaining boundary conditions are specified using a standard input dictionary. At the bottom and top of the domain (discrete phase inlet and outlet, respectively), as well as the lateral walls, the gradient of the electrical potential is set to zero by means of Equation (4).

∂ϕ ∂

( )

=0

Sf

Sf

(4)

f

Finite-volume discretization of the Laplacian (Equation (1)) and gradient (Equation (3) and (4)) schemes is completed using standard, second-order accurate Gaussian methods. An iterative preconditioned conjugate gradient (PCG) technique, as described by Ref. [8]; is employed to resolve the system of equations’ residual to a 10−12 criterion. Special considerations ensured that the described OpenFOAM solver could operate on a decomposed wire-mesh sensor domain and compute its signal response across multiple processors. A separate C++ postprocessing utility was established to reconsolidate decomposed data sets and generate a single wire-mesh sensor signal for each crossing point for a discrete time level. 3. Verification and validation Before the described finite-volume computational solver can be deployed in a predictive capacity, it is vital to assess the accuracy of the obtained numerical calculations and compare them to appropriate seminal research. To this end, a two-pronged verification and validation procedure is performed: (1) evaluate the congruence between results collected from the presented solution algorithm and identical data sets from Ref. [7]; (2) assess the various forms of numerical uncertainties (discretization, iteration, and round-off) and quantify their impact on the obtained signal responses. For this validation endeavor, a representative problem configuration is selected to homogenize such comparisons across computational solvers and mesh structures, the pertinent details of which are outlined in Table 1.

Fig. 3. Flowchart of OpenFOAM wire-mesh sensor electric potential solution algorithm.

potential field around wire-mesh sensors and their response to variable electrical conductivity fields. As such, the traversal of the discrete phase bubble does not depend on the pertinent physical forces (lift, drag, and buoyancy) or the interactive effect with the sensor electrodes. Nevertheless, as the presented solver is developed as an extension of OpenFOAM's CFD capabilities, it can be easily integrated into comprehensive multiphase models (volume of fluid, level set methods, etc.) which will handle the transport and physical deformation of the discrete phase. These advances are currently under development and will be summarized in subsequent investigatory research.

3.1. Model verification To ensure that the described OpenFOAM solver accurately predicts the signal response of a modeled wire-mesh sensor, the benchmark problem configuration outlined in Table 1 is analyzed and compared to identical computations performed by the routine developed by Ref. [7]. With the two finite-volume solvers operating in fundamentally different manners, specific considerations are made to certify parity in the compared data sets. In this regard, a dimensionless axial position (z ∗) is adopted (and defined in Equation (5)) to simplify the comparison of sensor signal responses.

2.2. Finite-volume model Beyond the development of the finite-volume solution algorithm, additional effort was put into generating high-quality, orthogonal computational grids that could be easily parameterized to evaluate multiple geometric wire-mesh sensor configurations. Using user-defined parameters for the geometric sensor factors, such as the number of transmitter and receiver electrodes, the axial and lateral electrode spacing (δax and δlat ), and axial and lateral boundary offset (Δax and Δlat ), an automated meshing procedure generates an input dictionary for the standard blockMesh utility. Each constructed finite-volume mesh has the same general blocking structure: two planes of wire-mesh sensor electrodes enclosed by three planes of uniform, orthogonal control volumes. An archetypical representation of the resultant finitevolume computational grid is displayed in Fig. 4. To simplify verification assessments against the work of [7]; the electrode shape profile can be generated in both cylindrical and rectangular configurations; both variants are depicted in Fig. 4. With the electric potential at the surface of transmitter electrodes

z∗ =

2z − H δax

(5)

z ∗,

the axial positions of the receiver Using the preceding definition for and transmitter electrode planes are conveniently located at z ∗ = −1.00 and z ∗ = +1.00 , respectively. Similarly, as the algorithm developed by Ref. [7] is inherently dimensionless in construction, and since the present investigation arbitrarily sets the active transmitter electrode potential to unity, an analogous nondimensionalization of the transmitted current is performed. At each discrete time level, the transmitted electric current (I ) at a receiver electrode is normalized by the corresponding calibration value (I0) obtained at the start of each simulation. As previously discussed, verification models constructed for the OpenFOAM solver feature square-profiled electrodes to allow a direct comparison with the numerical methodology of [7]. Verification simulations are performed at three discrete phase sizes (characterized by their spherical bubble radius relative to the axial pitch between the electrode planes) of rb δax = 1.00 , 2.00 , and 4.00 . Each 110

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Fig. 4. Computational mesh structures of wire-mesh sensor finite-volume model with cylindrical- (left) and square-shaped (right) electrodes. Table 1 Baseline wire-mesh sensor geometric parameters. Parameter

Symbol

Quantity

Wire-Mesh Sensor Size Electrode Diameter Superficial Bubble Velocity

nReceiver × nTransmitter dc vb

7×7 0.10 mm

Electrode Pitch, Axial Electrode Pitch, Lateral Boundary Distance, Axial Boundary Distance, Lateral

δax δlat Δax Δlat

0.10 mm s−1 1.50 mm 3.00 mm 10.00 mm 1.50 mm

of the analyzed spherical bubbles is released at the lateral center of the fluid column and advance axially through the centermost crossing point of the electrode network. Signal responses are monitored at four measurement sites, which are identified as locations A, B, C, and D in Fig. 5, by both potential field algorithms. Fig. 6 displays the average and maximum percent errors (ε ) between the predicted normalized transmitted currents for the analyzed bubble sizes at each of the observed crossing points. Overall, there is exceptional agreement between the constructed OpenFOAM algorithm and the finite-volume routine developed by Ref. [7]. A marginal increase in predictive error is observed with increasing discrete phase diameter, which is attributed to differences in discretization resolution (reduced bubble resolution at larger bubble sizes). With this satisfactory verification procedure, the presented OpenFOAM solver is deemed an effective predictive tool for predicting wire-mesh sensor behavior in bubbly flow topologies.

Fig. 5. Wire-mesh sensor crossing point identifiers and bubble size illustration.

domain into a finite series of control volumes, which serve to discretely approximate continuous partial differential equations. In addition to the conventional discretization uncertainty effects, the spatial resolution of the discrete phase bubble is also affected by the structure and size of the computational grid (a finer mesh will result in a more

3.2. Numerical uncertainty Of the three aforesaid forms of numerical uncertainty, the most adverse effects arise from the discretization of the computational 111

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Fig. 6. Average and maximum percent errors (ε ) between predicted signal responses from OpenFOAM solver and algorithm of [7].

[11]; is calculated for the outlined validation case. As the basis of the grid convergence index calculation, the present study employs three computational grids (with the structure displayed in Fig. 4). Employing uniform cell base sizes (h ) of 0.250 , 0.166, and 0.125 mm results in computational domains containing approximately 1.06, 3.59, and 8.50 million control volumes, respectively. Using the electric current transmitted to the receiver electrodes as the system response quantity (SRQ) of interest, the GCI can be computed for each discrete phase position via Equation (6):

GCII =

F I3 − I2 r23p − 1

(6)

where F is an empirical factor of safety, p is the observed order of convergence, and r is the mesh refinement factor between consecutive computational grids. Due to computational constraints and the configuration of the finite-volume model, a non-constant refinement factor was adopted, which resulted in the transcendental expression shown in Equation (7) for the observed convergence rate.

I3 − I2 I −I = r12 ⎜⎛ 2p 1 ⎞⎟ r23p − 1 ⎝ r12 − 1 ⎠

( = 1.00).

Fig. 7. Calibrated sensor response at location A for coarse

(

h

δax

)

= 9 , and fine

(

h

δax

)

= 12 density meshes

(

rb

δax

h

δax

)

(7)

For clarity, the subscripts present in Equations (6) and (7), 1, 2 , and 3, represent the corresponding values obtained from the coarse, medium, and fine density meshes, respectively. Likewise, paired subscripts (12 and 23) correspond to the relative grid refinement between the corresponding computational mesh densities. Fig. 7 illustrates the effects of discretization uncertainty in the sample problem outlined in Table 1 for the three described computational meshes (identified by their cell base size relative to the axial pitch between electrode planes). Based upon the normalized transmitted current at the centermost crossing point, which is shown on the

= 6 , medium

nominal bubble shape). Therefore, it is imperative to ensure the convergence of the numerical results with decreasing grid size and quantify the associated uncertainty at a given computational cost (number of control volumes). To quantify the discretization uncertainty present in the finite-volume solver and constructed models, the grid convergence index (GCI) of [9,10]; with practical modifications proposed by Ref. 112

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vertical axis of Fig. 7, satisfactory asymptotic convergence, particularly with respect to the medium and finest density grids, is observed. Using the outlined GCI methodology, the relative discretization uncertainty is computed for each discrete phase position. Along the centermost crossing point signal, the computed maximum relative GCI was 1.09% with the average relative discretization uncertainty limited to 0.09%. The quantified spatial uncertainty aligns with the observable invariance present in Fig. 7, indicating that results obtained from the OpenFOAM solution algorithm are both convergent with decreasing cell size and largely independent of grid density beyond the analyzed domains. Based on these uncertainty levels, ensuing simulations will utilize the medium density mesh arrangement and base cell size h δax = 9 to minimize the total computational requirements of the present study. Efforts are also made to minimize the effects of iteration and roundoff uncertainty on the obtained simulation results. With regards to iteration uncertainty, each electric potential field realization is subject to stringent convergence criteria with computations only advancing after the average Equation (1) residuals fall below 10−12. Additionally, both the base functionality and modified wire-mesh sensor solver employ a double-precision floating-point format, thus minimizing possible truncation errors present in the solution SRQs. By taking the aforementioned precautions, it is anticipated that the adverse effects of iteration and round-off uncertainties are negligible compared to discretization uncertainty, an assumption that has been found satisfactory for other computational calculations [11].

(

)

Fig. 8. Axial boundary distance (Δax ) effects on transmitted current magnitudes.

4.2. Axial domain size 4. Modeling considerations With insulating electrical conditions imposed at the entrance and exit of the computational stagnant fluid column, an additional parametric study is required to guarantee obtained responses are independent of boundary distance. Fig. 8 summarizes the parametric simulation results with the minimum calibrated current presented across several axial domain boundary distances (Δax ) normalized by the electrode diameter (dc ) . In Fig. 8, the vertical axis represents the normalized transmitted current from the central crossing point at the instance of bubble intersection for each of the investigated domain sizes. Additionally, each of the minimum current values is paired with its corresponding grid convergence index to demonstrate the statistical significance of the domain independence study. Based on the asymptotic behavior of the minimum transmitted currents, the effects of the axial boundaries vanish beyond Δax dc = 100 . For cases in which the domain extent is smaller than this required minimum, electric flux is reflected by the axial boundaries back into the measurement voxel, thereby incorrectly strengthening the perceived electric transmission. To remove these artifacts, domain lengths in the present investigation are sufficiently large as to not experience these reflection effects.

Beyond the verification and validation of the described finite-volume solution algorithm, the underlying geometrical assumptions of the constructed model, specifically the profile of the wire-mesh sensor electrode and distance between the electrodes and the insulated surfaces at the entrance and exit of the stagnant fluid column (Δax ), need to be assessed in terms of their impact on the transmitted current SRQ. 4.1. Electrode shape profile As one of the principal assumptions of their work [7], treated each nominally cylindrical wire-mesh sensor electrode as a linear assembly of cubic finite-volume cells. To assess the validity of such a simplification, three test cases are devised which employ a square electrode profile (as established in Fig. 4) for comparison against equivalent responses from the benchmark configuration featuring cylindrical electrodes. The following relationships between the square electrode diameter (ds ) and cylindrical (dc ) are investigated: 1. Square profile inscribed within cylinder ( 2 ds = dc ) . 2. Equal electrode surface area

(

4ds

π

)

= dc .

5. Computational results

3. Cylindrical profile inscribed within square (ds = dc ) . 5.1. Effect of bubble diameter Signal responses from the first and second square electrode layouts were nearly indistinguishable from equivalent cylindrical simulations. For both the inscribed profile and identical surface area configurations, incongruities between square and cylindrical electrode responses were well below discretization uncertainty levels, indicating that the results are largely invariant to the shape profile. However, for the scenario in which the square and cylindrical electrodes have an equivalent outer diameter, the signal response for the square formation is markedly lower (∼5%) as the bubble approaches the axial center of the crossing point. As such, it is recommended that wire-mesh sensor simulations employing the square electrode simplification of [7] ensure an equivalent surface area to the modeled cylindrical geometry. Results from the present investigation will not feature such simplifications; each of the constructed finite-volume models will explicitly resolve the curvature of the cylindrical-shaped electrodes.

With their superior spatial resolution relative to other experimental techniques, wire-mesh sensors, in conjunction with post-processing routines, are typically used to characterize geometric information about the discrete phase (diameter, volume, etc.). In this regard, the wiremesh sensor configuration defined in Table 1 is subjected to idealized spherical bubbles of varying diameters. For each of the six simulated diameters, the lateral center of the spherical bubbles is held invariant at the centermost crossing point, location A in the diagram present in Fig. 5. Calibrated signals for each of the investigated spherical bubbles, identified by their radius relative to the axial pitch between the electrode planes, are displayed in Fig. 9. Unsurprisingly, an inverse proportionality between discrete phase size and normalized signal response magnitude is readily observed for all of the analyzed scenarios. For both the rb δax = 0.25 and rb δax = 0.50 spherical bubbles, a unique 113

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Fig. 10. Calibrated transmitted current at central (A) and neighboring crossing

Fig. 9. Calibrated wire-mesh sensor signal response to variable-sized spherical bubbles passing through location A.

points (B, C, and D) around laterally-centered spherical bubble

(

rb

δax

)

= 1.00 .

through the insulating discrete phase; or (2) along the periphery of the bubble surface in the highly conductive continuous medium. Because the boundaries of the computational domain are treated with insulating conditions, a higher electric current density, caused by the asymmetric electric potential field and amplification of electric field intensity in these conductive areas, passes around the large diameter bubble to the receiver electrode. Therefore, the current signal computed from this idealized configuration will never fully reach zero, but rather asymptotically approach it with increasing bubble size.

local maximum in the calibrated current signal appears at the axial center of the electrode planes. As the spherical bubbles begin to interact with the receiver electrode (z ∗ = −1.50 and z ∗ = −2.00 for rb δax = 0.25 and rb δax = 0.50 , respectively), the typical decline in current is observed. However, unlike the larger diameter spheres, the normalized current signal begins to rise as the bubbles cross into the interstitial region between the electrode planes. To explain this process, one must consider the magnitude of the electromotive force in the highly nonlinear electrical conductivity field produced by the presence of the discrete phase. For a wire-mesh sensor surrounded by a uniform continuous medium, a symmetric electric potential field forms between the activated transmitter electrodes and the adjacent receivers. As the traveling bubble distorts the local electrical conductivity and potential fields, the magnitude of the electric field vector (E = −∇V ) , the driving force of electric transmission, increases within the conducting region at the interface of the discrete and continuous phases. Because the rb δax = 0.25 bubble is smaller than the lateral extent of the sensitive volume at the inspected crossing point, and such an insignificant portion of the two electrodes is insulated by the spherical caps of the rb δax = 0.50 discrete phase, a net increase in transmitted current is observed in each of the calibrated signals. Conversely, this effect is absent from the other, larger diameter bubbles because the increase in electric field intensity occurs outside the highly sensitive region within the measurement voxel. Electric potential field symmetry, and lack thereof in the proximity of an inhomogeneous medium, can also be used to characterize the signal response to larger volume bubbles. When large volume bubbles pass through the geometric center of the wire-mesh sensor (location A), the entire measurement voxel is saturated with an insulating medium; therefore, the signal response should tend towards zero. Yet, the transmitted currents for larger diameter bubbles in Fig. 9 does not coincide with this fundamental premise. Illumination of this apparent contradiction can occur by invoking a simple, parallel two-branch electric circuit between an active transmitter electrode and each of the receivers. Under normal calibration conditions (i.e., full immersion within a highly conductive continuous phase), current travels along the potential gradient via the least electrically resistive path, which is directly across the measurement voxel to the receiver plane. When larger diameter bubbles envelop a portion of the active transmitter electrode, the parallel electric circuit forms, offering two routes for current to traverse the potential gradient: (1) via the shortest physical distance

5.2. Neighboring crossing point behavior As previously discussed, wire-mesh signal response at a given crossing point is dominated by the phasic composition within the measurement voxel – i.e., where the electrical conductance between the electrodes is maximized. However, current traveling between the two electrode planes is not limited to direct passage across the measurement site, especially in the presence of a highly maldistributed electric potential field. To illustrate this phenomenon, Fig. 10 displays the signal response at the periphery crossing positions (locations B, C, and D in Fig. 5) for the baseline sensor configuration outlined in Table 1. For this inspection, a 3 mm diameter spherical bubble rb δax = 1.00 traverses vertically through the stagnant continuous medium while laterally centered at the middle crossing point (location A). Based on the diameter of the bubble, and the lateral pitch of the wire-mesh sensor electrodes, a signal response should occur only at location A where the discrete phase fully encapsulates the measurement voxel. Despite these initial intuitions, profound responses are seen at the neighboring crossing points with observable overshoots as the bubble begins to contact the receiver electrode. Interestingly, the most intense overshoot in the current signal (about 3% above the calibrated condition) occurs at location B. This occurrence was first recognized by Ref. [7] where it was linked to the broken potential field symmetry in the vicinity of inhomogeneous conductivity mediums. This hypothesis is confirmed via careful observation of the computed electric potential fields, which are displayed in Fig. 11 for both the calibration condition (continuous phase only) and position of maximum overshoot at location B (z ∗ = 2.00) . As the discrete phase progresses into the measurement voxel between the centrally located transmitter and receiver electrodes, the electric potential field compresses around the bubble, causing an

(

114

)

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Fig. 11. Contours of electric potential field for calibration conditions (left) and around a

rb

δax

= 1.00 spherical bubble (z ∗ = −2.00) .

increase in electric field strength at the surface of the neighboring receivers, which in turn leads to the overshoot behavior demonstrated in Fig. 10. 5.3. Parametric lateral crossing position In the previous discussion regarding signal responses of neighboring receiver electrodes, it is established that electric current traveling between the two electrode planes is not limited to direct passage across the measurement voxel. Within those analyses, symmetry of the electric potential field is distorted due to the passage of the discrete phase bubble directly through the lateral center of the highly sensitive region between the electrode planes. To expand the understanding of wiremesh sensor behavior, an additional study is conducted to describe the signal response beyond that of laterally constrained spherical bubbles. Using the reference sensor geometry outlined in Table 1, electric potential field calculations are performed at intermediate bubble positions between the central crossing point (location A) and adjacent lateral electrodes. Five different locations, defined incrementally by distance from the lateral column center in Fig. 12, are investigated and related to the baseline configuration. To provide parity across this parametric analysis, the discrete phase is held invariant (3 mm spherical bubble) at each of the prescribed lateral positions. The predicted signal response at the central crossing point for each bubble arrangement is summarized in Fig. 13, which reveals an indirect proportionality between the distance from the measurement site and the decrement in transmitted electric current. As expected, the normalized current signal predicted at the central crossing point, location A, has the most significant response to the bubble traveling directly through the sensitive volume. With increasing lateral bubble distance from location A, i.e., less discrete phase contact with the measurement voxel, an increase in transmittance is observed, leading to an increase in the predicted electric current values. Of particular note, the axial position of the minimum transmitted current is not constant at z ∗ = 0 like the other presented computations. Like the behavior of the neighboring crossing point analysis, the electric potential field symmetry of the calibration condition is broken by the discrete phase. These asymmetries influence not only the magnitude of the electric current (and thus perceived size of the bubble), but also the

Fig. 12. Intermediate lateral discrete phase positioning within standard wiremesh sensor geometry (subset of Fig. 5).

apparent position of the bubble as it progresses through the sensor. Thus, more comprehensive calibration methodologies, such as those adopted by Ref. [7]; are paramount to the development of trustworthy experimental data sets collected from wire-mesh sensors. 5.4. Spheroidal bubble shape effects While parametric studies of spherical discrete phase size and lateral positioning aids in the understanding of wire-mesh sensor behavior in bubbly flow regimes, the assumption of idealized spherical bubbles may not necessarily correspond to realistic nuclear flow topologies. As a volume of discrete phase rises through a stagnant fluid column akin to the canonical case presented in this theoretical investigation, its shape would morph and diverge from the idealized spheres modeled to this point. This change in size and structure would, in turn, have some quantifiable effect on the electric potential field as the bubble progresses through the measurement voxel. Although not the focus of the presented OpenFOAM solver, the effect of spheroidal and generic ellipsoidal discrete phases can be preliminarily analyzed. In this manner, 115

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signal at the axial center of location A. To reiterate, this feature is caused by the enhancement of the electric field strength within the sensitive volume not occupied by the insulating discrete phase. As the geometric centroid of the discrete phase coincides with the axial center of location A, the bubble volume is not large enough to occupy the entirety of the sensitive volume, thus leading to the rise in transmitted current relative to the calibration case. Because of their reduced volume relative to their spherical counterpart, the influence of the spheroidal bubbles on the transmitted current signals occurs slightly farther downstream (z ∗ = −2.50 vs. z ∗ = −3.00 for the spherically shaped variants). 6. Conclusions Finite-volume computations of the electric potential field around a wire-mesh sensor are evaluated using a custom OpenFOAM solution algorithm. A comprehensive solution verification and validation procedure is conducted using the seminal work of [7] as the basis of comparison. To justify the geometric simplifications and modeling assumptions, a detailed investigation and discussion of the various numerical considerations is provided along with their relative impact on the sensor's response to an insulating discrete phase bubbles. Parametric studies of spherical diameter, lateral crossing position, and spheroidal shape factor are executed to analyze their significance with respect to the transmitted current between the electrode planes. Variations in the local electrical conductivity within the measurement voxel between electrode planes have a profound effect on the transmitted electric current as the resulting asymmetrical electric potential field departs from calibration conditions. Observed overshoots in the calibrated current signals are linked back to the maldistributed potential field around the electrically insulating discrete phase and enhancement of electric field strength at the bubble boundaries. Serving as a natural evolution of [7]; the present study establishes a comprehensive modeling framework with which to perform potential field computations within an arbitrarily configured wire-mesh sensor. Because of its modular nature, the described algorithm is easily integrated with OpenFOAM's multiphase models, allowing future exploration into the interaction of bubbles with the wire-mesh sensor electrode surfaces. The validated computational methodology presented herein represents the crucial first step towards that goal, which can provide additional insight into wire-mesh sensor operation and permit the development of a more physics-based calibration approach for experimental WMS data sets.

Fig. 13. Calibrated signal response at location A to spherical bubble

(

rb

δax

)

= 1.00 at intermediate lateral positions.

Acknowledgements This research was performed using funding received from the United States Department of Energy's Nuclear Energy University Program (NEUP) Fellowship (13-5726). The authors would also like to thank Texas A&M University's High Performance Research Computing (HPRC) group for providing the necessary computational resources for this numerical investigation.

Fig. 14. Spheroidal (a = b = 1.50 mm) shape influence at location A.

the discrete phase radius in the z -direction (which is governed by the bubble parameter c ) is modified to generate spheroidal bubbles. Bubbles in this parametric investigation are held uniform in the x - and y -directions (a = b = 1.50 mm) as the axial parameter is varied. For each investigated spheroid shape, here characterized by the radius constraint in the z -direction relative to that of the x -direction, calibrated wire-mesh sensor signals are presented in Fig. 14. The response to the standard spherical rb δax = 1.00 bubble, which is defined by the c = 1.000 curve, is included in Fig. 14 for easier comparison. Similar to a the effects witnesses during the variable diameter study, an inverse proportionality between discrete phase volume and transmitted current is plainly discernible. Much like the rb δax = 0.25 and 0.50 spherical bubble responses presented in Fig. 9, the three spheroidal bubbles (c a = 0.125, 0.250 , and 0.500 ) possess a distinctive local maximum in the calibrated current

Appendix A. Supplementary data Supplementary data to this article can be found online at https:// doi.org/10.1016/j.flowmeasinst.2019.04.009. References [1] R.A. Berry, J.W. Peterson, H. Zhang, R.C. Martineau, H. Zhao, L. Zou, D. Andrs, RELAP-7 Theory Manual, Idaho National Laboratory, Idaho Falls, Idaho, 2014, p. 154. [2] L. Zou, R.A. Berry, R.C. Martineau, D. Andrs, H. Zhang, J.E. Hansel, J.P. Sharpe, R.C. Johns, RELAP-7 Closure Correlations, Idaho National Laboratory, Idaho Falls, Idaho, 2017, p. 60. [3] C.L. Smith, Y.-J. Choi, L. Zou, RELAP-7 Software Verification and Validation Plan,

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Idaho National Laboratory, Idaho Falls, Idaho, 2014, p. 160. [4] Johnson, I. D., 1987. Method and Apparatus for Measuring Water in Crude Oil. United States Patent, No 4,644,263. [5] H.-M. Prasser, A. Böttger, J. Zschau, A new electrode-mesh tomograph for gas-liquid flows, Flow Meas. Instrum. 9 (1998) 111–119. [6] C. Tompkins, H.-M. Prasser, M. Corradini, Wire-mesh sensors: a review of methods and uncertainty in multiphase flows relative to other measurement techniques, Nucl. Eng. Des. 337 (2018) 205–220. [7] H.-M. Prasser, R. Häfeli, Signal response of wire-mesh sensors to an idealized bubbly flow, Nucl. Eng. Des. 336 (2018) 3–14. [8] F. Moukalled, L. Mangani, M. Darwish, The Finite Volume Method in Computational Fluid Dynamics: an Advanced Introduction with OpenFOAM and Matlab, Springer International Publishing, Cham, Switzerland, 2016. [9] P.J. Roache, Perspective: a method for uniform reporting of grid refinement studies, J. Fluids Eng. 116 (1994) 405–413. [10] I.B. Celik, U. Ghia, P.J. Roache, C.J. Freitas, H. Coleman, P.E. Raad, Procedure for estimation and reporting of uncertainty due to discretization in CFD applications, J. Fluids Eng. 130 (2008) 4. [11] W.L. Oberkampf, C.J. Roy, Verification and Validation in Scientific Computing, Cambridge University Press, 2010.

F : factor of safety GCI : grid convergence index h : mesh base size, mm H : column height, mm I : electric current, A J : electric current density, A mm−2 p : observed order of convergence r : grid refinement factor S : face area vector, mm2 v : velocity, mm s−1 z : axial position, mm

Nomenclature

Subscripts and Superscripts

Symbol

Symbol: Description ax : axial b : bubble c : cylindrical electrode f : face lat : lateral s : square electrode

Greek Symbols ε : percent error δ : electrode pitch, mm Δ : boundary distance, mm σ : electrical conductivity, mm−1 Ω −1 ϕ : electric potential, V

a : discrete phase radius, x -direction, mm A : electrode area, mm2 b : discrete phase radius, y -direction, mm c : discrete phase radius, z -direction, mm d : electrode diameter, mm E : electric field vector, V mm−1

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