TWO MODELS of ELECTRICAL IMPEDANCE for ELECTRODES with TAP WATER and THEIR CAPABILITY to RECORD GAS VOLUME FRACTION Revista Mexicana De Ingeniería Química, Vol

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Revista Mexicana de Ingeniería Química ISSN: 1665-2738 [email protected] Universidad Autónoma Metropolitana Unidad Iztapalapa México

Rodríguez-Sierra, J.C.; Soria, A. TWO MODELS OF ELECTRICAL IMPEDANCE FOR ELECTRODES WITH TAP WATER AND THEIR CAPABILITY TO RECORD GAS VOLUME FRACTION Revista Mexicana de Ingeniería Química, vol. 15, núm. 2, 2016, pp. 543-551 Universidad Autónoma Metropolitana Unidad Iztapalapa Distrito Federal, México

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CONTENIDO TWO MODELS OF ELECTRICAL IMPEDANCE FOR ELECTRODES WITH TAP WATER ANDVolumen THEIR 8, número CAPABILITY 3, 2009 / Volume TO 8, RECORD number 3, GAS2009 VOLUME FRACTION

DOS MODELOS DE IMPEDANCIA ELECTRICA´ PARA ELECTRODOS CON AGUA POTABLE Y SU CAPACIDAD DE REPRESENTAR LA FRACCION´ VOLUMEN DE 213 Derivation and application of the Stefan-MaxwellGAS equations * (Desarrollo y aplicaciónJ.C. de Rodr las ecuaciones´ıguez-Sierra de Stefan-Maxwell) and A. Soria Departamento de Ingenier´ıade Procesos e Hidráulica.DivisiónCBI, Universidad AutónomaMetropolitana-Iztapalapa. San Stephen Whitaker Rafael Atlixco No. 186 Col. Vicentina, CP 09340 Cd. de México,México. Received May 24, 2016; Accepted July 5, 2016 Biotecnología / Biotechnology

Abstract 245 Modelado de la biodegradación en biorreactores de lodos de hidrocarburos totales del petróleo Bubble columns are devices for simultaneous two-phase or three-phase flows. Phase interactions produce several flow patterns where the voidintemperizados fraction is an importanten suelos y variablesedimentos involved in the behavior and fundamental in flow pattern transitions. Electrical impedance sensors (EIS) determine void fraction and perform as fast response, passive elements, exhibiting (Biodegradation modeling of sludge bioreactors of total petroleum hydrocarbons weathering in soil resistive, capacitive and inductive behaviors highly dependent upon the excitation frequency. A simple electrical model frequently used is a setand of sediments) a resistance and a capacitor connected in parallel. Same elements can also be arranged in series, as we do here. We identifyS.A. Medina-Moreno, three behaviors S. inHuerta-Ochoa, the series and C.A. parallel Lucho-Constantino, arrangements, L. asAguilera-Vázquez, well as in the experimentalA. Jiménez- data. While the ones of theGonzález series arrangement y M. Gutiérrez-Rojas are coincident with experimental data, the ones of the parallel model are only partially coincident259 at Crecimiento, high frequencies. sobrevivencia Moreover, y adaptación while the de parallel Bifidobacterium model is insensitiveinfantis a condiciones to changes ácidas in gas volume fraction in the resistive range, the series model presents sensitivity to changes in the gas volume fraction. Therefore, on these grounds the series arrangement (Growth, survival exhibits and abetter adaptation performance of Bifidobacterium than the parallelinfantis to model. acidic conditions) Keywords: electricalL. impedance, Mayorga-Reyes, series P. model, Bustamante-Camilo, parallel model, A. tap-waterGutiérrez-Nava, model, E. voidBarranco-Florido fraction, air-water y A. Azaola- flow, bubble column. Espinosa Resumen 265 Statistical approach to optimization of ethanol fermentation by Saccharomyces cerevisiae in the Las columnas de burbujeo son dispositivos para el flujo simultaneo´ de dos o tres fases. La interaccion´ entre fases produce varios patrones de flujo,presence donde of laValfor® fracci on´zeolite volumen NaA de gas es una variable importante implicada en el comportamiento de los patrones de flujo y es fundamental en sus transiciones. Los sensores de impedancia electrica´ (SIE) determinan la (Optimización estadística de la fermentación etanólica de Saccharomyces cerevisiae en presencia de fraccion´ de vac´ıo y se desempenan˜ con respuesta rapida,´ elementos pasivos que despliegan comportamientos resistivo, capacitivo e inductivozeolita ampliamente Valfor® dependienteszeolite NaA) de la frecuencia de excitacion.´ Un modelo electrico´ sencillo que se ha usado con frecuencia esG. unInei-Shizukawa, conjunto de H. una A. resistenciaVelasco-Bedrán, y un condensadorG. F. Gutiérrez-López en paralelo. and H. Los Hernández-Sánchez mismos elementos pueden ser dispuestos en serie, como lo hacemos aqu´ı. Identificamos tres comportamientos de los arreglos en serie y en paralelo, as´ı como de los datosIngeniería experimentales. de procesos Los / del Process arreglo engineering en serie coinciden con los de los datos experimentales, en tanto que los del modelo en paralelo solo´ parcialmente para altas frecuencias. Ademas,´ mientras el modelo en paralelo es insensible a 271 Localización de una planta industrial: Revisión crítica y adecuación de los criterios empleados en los cambios de fraccion´ volumen de gas en el regimen´ resistivo, el modelo en serie presenta sensibilidad a los cambios en esta decisión la fraccion´ volumen de gas. Sobre estas bases, el arreglo en serie presenta un mejor desempeno˜ que el modelo en paralelo. Palabras clave: impedancia (Plant site electrica,´ selection: modelo Critical en serie,review modelo and adequation en paralelo, criteria modelo used in del this agua, decision) fracci on´ vac´ıo, flujo aire-agua, columna de burbujeo. J.R. Medina, R.L. Romero y G.A. Pérez

1 Introduction columns, where the liquid is the continuous phase and the gas is thinly dispersed in small bubbles, to misty or fog flows. In these flows the gas is the Dispersed multiphase flows play an important role continuous phase and the liquid is dispersed in tiny in the process industries. Among these flows, those drops or fog. A third phase can also be present with two or three phases are the most frequent. Gas- in a bubble column as a dispersed phase; thus, a liquid flows inside pipes are used for many industrial solid powder or swarms of solid particles or pellets applications (Falcone, 2009a) spanning from bubble *Correspondingn author. E-mail: [email protected] Tel. 55-5804-4648, Fax 55-5804-4900

Publicado por la Academia Mexicana de Investigacion´ y Docencia en Ingenier´ıa Qu´ımica A.C. 543 Rodr´ıguez-Sierra and Soria. Revista Mexicana deIngenier ´ıaQu ´ımica Vol. 15, No. 2 (2016) 543-551

can configure the third phase, as in waste water imaginary part is the reactance. Impedance multiphase treatment columns or in three-phase fluidized bed flow measurements can be divided in two broad chemical reactors. There is interest on studying the topics: the impedance intrusive device, with electrodes occurrence of several geometrical configurations of separated by few millimeters, commonly mounted on the dispersed phases, called flow patterns, in a bubble an immersion probe and used for local measurements column, both for design and for operation purposes. of the void fraction and the impedance not-intrusive Some parameters, such as the physical properties of device, flushed to the inside wall or at the outside wall the materials, their mass flow rates, their volume of the pipe, used for the whole cross sectional void fractions, the column length and diameter, as well fraction (Micaelli, 1982; Bernier, 1982; Tournaire, as its tilting, determine not only the flow patterns 1986, 1987). but also the transitions between them (Taitel et al., Studies on air-water flows using electrical 1980; Spedding and Nguyen, 1980; Li et al., 2016). impedance are usually performed to measure Some methods have been proposed to classify flow impedance as a function of the frequency and the patterns using characteristic variables such as void voltage or the intensity of the applied current (Ko fraction fluctuations or differential pressure, which are et al., 2015). To measure the void fraction, sensors supposed to reflect the flow configuration (Song et al. are made with electrodes or sets of electrodes. Each 1995; Costigan and Whalley 1997; Cheng et al. 2002; electrode embraces a detection volume crossed by the Rodr´ıguez, 2006). Transitions between flow patterns air-water flow. Volumetric fractions and interfacial are complex phenomena closely related to interactions areas are the more important geometrical parameters between bubbles and fluid, where the void fraction in multiphase flows. In general, both are independent (the gas volume fraction) becomes a very important parameters (Soria and de Lasa, 1992). Bernier (1982), dynamic variable (Zenit and Hunt, 2000; Banasiak following Maxwell, establishes relationships between et al., 2014) whose stability analysis can enlighten electrical parameters and the volume fraction of the the occurrence of the regime transitions (Soria and conductive phase. Thus, relationships with interfacial Salinas-Rodr´ıguez, 2013; Sanchez-L´ opez´ et al. 2011; areas should be established specifically, probably for Cheng et al. 2002). different arrangements of small electrodes and taking Various definitions are used for specifying the care about the geometrical shapes, as Maxwell theory void fraction: local-instantaneous, time averaged, is being applied. cross-sectional and volumetric among others. The Sensors with specific characteristics of electrodes volumetric void fraction is based on the relative (geometry, electrode size, electrode shape), for volume occupied by the respective phase. In this several operating conditions (excitation frequency, study, the local volume fraction of a given fluid, at conductance method or capacitance method) have a given point x in a two-phase mixture, is defined as been developed (Bernier, 1982; George et al., 2001; the volume occupied by the fluid in a representative Devia and Fossa, 2003, Yang et al., 2003). Tournaire elementary volume (REV), as a fraction of the whole (1986) defined the geometry of the probe by the width REV. The centroid of the REV is placed at point and the number of electrodes. x. In particular, the REV in the experimental EIS The air-tap water two-phase system, flowing (c. fr. Fig. 3) is coincident with the slice sensed through the detection zone, is usually modeled by the electrodes. There are some volume fraction by a capacitor and a resistance. The elementary experimental measurement techniques (Prasser, 2007; electrical model of the impedance sensor and the Falcone, 2009b; Coutinho et al. 2014) such measuring system that operates without electrolysis as the acoustic attenuation, microwave attenuation, near electrode surfaces can be compared to a RC ultrasound, tomography, wire-mesh and electrical circuit (Rocha, 2008; Mansfeld et al., 1993). The impedance. This last technique is the simplest and analysis of a RC model shows a simple way to probably the cheapest of all techniques (Rocha, 2008) associate the overall two-phase mixture behavior with and allows measurements of the void fraction with their corresponding arranged electrical elements. The 3 resolution times of 10− s and smaller. parallel arrangement seems to have been frequently The electrical impedance is frequently used assumed without experimental verification (dos Reis for the characterization of components, systems, and da Silva, 2014; Micaelli, 1982; Bernier, 1982; and materials (Mart´ınez Gomez´ and Soria, 2003). Tournaire, 1986, 1987, Sainz-Jabardo and Boure,´ Electrical impedance is expressed as a complex 1988; Kytomaa¨ and Brennen, 1988; Jaworek et al., number, whose real part is the resistance and whose 2004; Tri, 2005; Wu et al., 2015), series arrangement

544 www.rmiq.org Rodr´ıguez-Sierra and Soria. Revista Mexicana deIngenier ´ıaQu ´ımica Vol. 15, No. 2 (2016) 543-551

does not seem to have been reported. This kind of models is used for the evaluation of the behavior of the electrical impedance sensor in the frequency domain, to determine the operation mode, i.e. the capacitive mode or the resistive mode. The purpose of this communication is to show that the series model of the basic capacitor/resistance arrangement of electrodes in the air-tap water two- phase systems in a bubble column ﬁts the experimental data in a wider interval of excitation frequencies than the parallel model, including the low range frequencies, where electrode capacitive behavior Fig. 1. Voltage divider with two components arranged is known to occur. It is also our purpose to in series. show that the series model provides a simple, yet meaningful piece of information for further modeling We analyzed the case where Z1 is purely resistive development, since it discriminates between diﬀerent and Z2 is the multiphase system composed by both volume fraction values in the resistive range. resistive and capacitive elements (see Fig. 2), therefore Z2 is a function of ω j. Substituting Z1 = R and Z2 = Z(ω j) the previous expression gives:

2 Transfer functions of two-phase V1 R H1(ω j) = = (2) system Vin R + Z(ω j) and

A voltage divider is set with respect to ground by V2 Z(ω j) connecting two electrical impedances in series, as H2(ω j) = = (3) Vin R + Z(ω j) shown in Fig. 1. The input voltage is applied across where, H (ω j) is the transfer function according to R the series impedances and the output is the voltage 1 and H (ω j) is the transfer function according to Z(ω j). across Z or Z . The impedances Z and Z may 2 2 1 1 2 The multiphase system is represented by Z(ω j), be arranged in any combination of elements such as but the configuration of the arrangement between resistors, inductors, and capacitors. Table 1 shows the resistive and capacitive parts is not yet determined. impedance elements. Because of the electrical properties of tap-water, the Applying Ohm’s law, the relationship between impedance of the two-phase system is made up by input voltage, V , and the output voltage, V or V , in 1 2 a capacitance CTP and a resistance RTP distributed can be found and the transfer function or divider’s over the detection volume, if volume is defined by voltage ratio for this circuit is: the lines of an electric field associated to the electrode system. The two possible configurations of Z(ω j) to Vi Zi be tested are: (1) the components connected in series Hi = = (1) Vin Z1 + Z2 Zs(ω j) (Fig. 2(a)) and (2) the components connected in parallel Zp(ω j) (Fig. 2(b)). Here the subscripts s where, the subscript “i ”denotes 1 or 2, according to and p stand for the series and parallel arrangements, the point of measure. respectively. These impedance expressions for the series and parallel RC circuits are: 1 Table 1. Impedances of passive elements. Zs(ω j) = RTP + (4) CTPω j Element Impedance and R(Resistor) Z = R R z Z (ω j) = TP (5) C(Capacitor) Z = 1 p + ω jωCz 1 CTPRTP j L(Inductor) Z = jωLz The relationship between the impedance of water and the impedance of the two-phase flow system is

www.rmiq.org 545 J.C. Rodr´ıguezy A. Soria J.C. Rodr´ıguezy A. Soria

V2 Z(ω j) H2(ω j) =V = Z(ω j) (3) 2Vin R + Z(ω j) H2(ω j) = = (3) Vin R + Z(ω j) 133 where, H1(ω j) is the transfer function according to R and H2(ω j) is the transfer function according to Z(ω j). 134133 where,TheH multiphase1(ω j) is the system transfer is function represented according by Z( toω jR),and butH the2(ω configurationj) is the transfer of the function arrangement according between to Z(ω resistivej). and 134 ω 135 capacitiveThe multiphase parts is not system yet determined. is represented Because by Z( j of), butthe the electrical configuration properties of the of arrangement tap-water, the between impedance resistive of the and two- 135 capacitive parts is not yet determined. Because of the electrical properties of tap-water, the impedance of the two- 136 phase system is made up by a capacitance CTP and a resistance RTP distributed over the detection volume, if volume 136 phase system is made up by a capacitance C and a resistance R distributed over the detection volume, if volume 137 is defined by the lines of an electric fieldTP associated to the electrodeTP system. The two possible configurations of 137 is defined by the lines of an electric field associated to the electrode system. The two possible configurations of 138 Z(ω j) to be tested are: (1) the components connected in series Zs(ω j) (Fig. 2(a)) and (2) the components connected 138 Z(ω j) to be tested are: (1) the components connected in series Zs(ω j) (Fig. 2(a)) and (2) the components connected 139 in parallel Zp(ω j) (Fig. 2(b)). HereRodr´ıguez-Sierra the subscripts s and andp stand Soria. forRevista the series Mexicana and parallel deIngenier arrangements,´ıaQu ´ımica respectively. Vol. 15, No. 2 (2016) 543-551 139 in parallel Zp(ω j) (Fig. 2(b)). Here the subscripts s and p stand for the series and parallel arrangements, respectively.

and R + RR C jω H p( jω) = TP TP | 1 | R + R + RR C jω TP TP TP 2 2 [ωRRTPCTP] + R = (11) 2 2 [ωpRRTPCTP] + [R + RTP] and similarly, forp point 2:

1 + jωR C Hs( jω) = TP TP | 2 | jω(R + R )C + 1 TP TP ω ω 2 (a)(a) Impedance ImpedanceZsZ(ωs(j)j) with with (b)(b) Impedance ImpedanceZp(Zωpj() withj) with [ωRTPCTP] + 1 components connected in components connected in = components connected in components connected in 2 (12) seriesseries parallelparallel [ω(R + RTP)CTP] + 1

Fig.Fig. 2: 2: Resistor Resistor/impedance/impedance arrangements arrangements in in voltage voltage divider. divider. and Fig. 2. Resistor/impedance arrangements in voltage 140140 RTP divider. H p( jω) = 141141 These impedance expressions for the series and parallel RC circuits are: 2 These impedance expressions for the series and parallel RC circuits are: | | R + RTP + RTPCTPω j

developed by Bernier (1982) after1 Maxwell.1 The two- RTP Zs(ωωj) = RTP + = (4) (13) phase resistanceZRs(TPj)and= R theTP + two-phase capacitance (4) 2 2 CTPCTPω jω j [ωRRTPCTP] + [R + RTP] CTP are given by: 142 and p 142 and R Now the frequency response for each detection ω = RTPRTP ZZp( (ωj)j) = z point can be plotted(5) (5) if the values of Rz, Cz and α are p RTP =1 + CTPRTPω j (6) 1 + C3TPα RTPω j 1 +α known. 143 The relationship between the impedance of water and the impedance2 of the two-phase flow system is developed by 143 The relationship between the impedance of water and the− impedance of the two-phase flow system is developed by 144 Bernier (1982) after Maxwell. The two-phase resistance R TP and the two-phase capacitance CTP are given by: 144 Bernier (1982) after Maxwell.and The two-phase resistance RTP and the two-phase capacitance CTP are given by: Rz 3α 3 Experimental technique CTPRTP==Cz 1 Rz (7) (6) RTP =1 −32α + α (6) −1 2+α3α ! − 2+α 3.1 Experimental facility set-up where R and C are resistance and capacitance values 145 and z z 145 and for water without bubbles (α = 0)3α to be found by fitting Figure 3 is a schematics of the facility, as designed for CTP = Cz 1 3α (7) experimental points.CTP = Cz −1 2 + α this study. This consists(7) of a function generator and a − 2 +!α resistance, the test section, and an oscilloscope. The 146 where Rz and Cz are resistance andThe capacitance transfer functions values for can water now without be! expressed bubbles for(α = 0) to be found by fitting 146 where Rz and Cz are resistance and capacitance values for water without bubbles (α = 0)test tobe section found used by fitting in the experiments is a cylindrical 147 experimental points. models in series and in parallel according to the 147 experimental points. pipe made of transparent Plexiglasrwith an inside 148 The transfer functions candetection now be expressed points 1 for and models 2. Substituting in series and Eq. in parallel(4) and according Eq. to the detection points 148 The transfer functions can now be expressed for models in series and in parallel according to the detection. points . 149 1 and 2. Substituting Eq. (4) and(5) Eq.into (5) Eq. into (2), Eq. we (2), get, we at get, the at detectionthe detection point point 1, 1, for for seriesdiameter model: of 0 053m and a height of about 0 5m. The 149 1 and 2. Substituting Eq. (4)series and Eq. model: (5) into Eq. (2), we get, at the detection point 1, for seriestest sectionmodel: is filled with tap water at ambient pressure jωRC Hs( jω) = TP (8) 1 s jωRCTP Hs ( jω) =1 + (R +jωRRCTP)TPCTP jω (8) H (1jω) = + + ω (8) 1 11+ (R(R+ RRTP)C)CTPjωj Manuscrito sometido a la Revista MexicanaTP TP deIngenier´ıaQu´ımica 5 Manuscritoand for parallel sometido model a la Revista Mexicana deIngenier´ıaQu´ımica 5

p R + RRTPCTP jω H1 ( jω) = . (9) R + RTP + RRTPCTP jω

Finding their modulus, the previous expressions give:

jωRC Hs( jω) = TP | 1 | 1 + (R + R )C jω TP TP RC ω = TP (10) 2 [ω(R + RTP)CTP] + 1 Fig. 3. Schematics of the experimental facility. p 546 www.rmiq.org Rodr´ıguez-Sierra and Soria. Revista Mexicana deIngenier ´ıaQu ´ımica Vol. 15, No. 2 (2016) 543-551

and temperature. The EIS has two sections of stainless using the series model, this will be discussed in the steel plates. Steel plates are assembled, alternating next sections. The optimization is performed using with insulator plates 0.5mm thick. Each electrode is least squares in two stages. In the first stage optimal composed of an active pole, made up by a selected value of Rz was found only in the resistive interval, number of metal plates, diametrically opposed to a this step is justified by the specific knowledge that the common pole in each section and spanning 90 degree transfer function is a constant in this interval, finding arcs on the circumference of the pipe, as can be directly the value of Rz. In the second stage the value observed in Fig. 3. Each section has 50 metal of the capacitance Cz was found by least squares. plates flush mounted to the inner wall, with 1mm axial In Fig. 4(a) the frequency response of both length. The opposed common pole in each section models, in a linear plot, according to detection point is 75mm in axial length. Fig. 3 shows the electrode 1 is depicted. The Hs(α = 0) plot shows that the configuration. series system has an inductive behavior at frequencies between 0.1 and 1 Hz, whereas above 1 kHz it keeps 3.2 Experimental procedure a resistive behavior. The Hp(α = 0) plot shows that the parallel model has two resistive behaviors at To generate a single electrode, all the metal plates are low and high frequencies, whereas, in the middle, connected to a single cable, thus we get the maximum an inductive behavior about 1 to 100 Hz is apparent. size electrode. A sinusoidal a.c. signal is selected Same frequency response is shown in decibels (dB) to avoid polarization and reaction in the electrodes in Fig. 4(b). It is apparent that at small frequencies (d.c.=0 volt) and a 5-volt potential peak to peak is (1) the curvature of series model is smoother than the applied across the 1kΩ resistance and the EIS. This curvature in linear scale and (2) The two horizontal signal is provided by the function generator Insteck lines of the parallel model are closer to each other as (GFG-3015), where the frequency is varied at four well as very close to zero. points per decade in the range of 0.1 Hz to 1 MHz; this In Fig. 5(a) the frequency response of models is 1,2.5,5,7.5,10,25,50,75,100, Hz...,1 MHz. The according to detection point 2 is depicted in linear experimental transfer function can be obtained by scale. The Hs(α = 0) linear plot shows that the measuring the input voltage Vin and the output voltage series system has a resistive behavior above 1 kHz, V1 or V2, for each one of the selected points, with the while a capacitive behavior about 0.3 Hz to 3 Hz is oscilloscope (Tektronix TDS 1002) and replacing the apparent. In addition, the Hp(α = 0) linear plot for the experimental values in Eq. (2) or (3) according to the parallel system presents a behavior with a resistive low case. frequency range, below 1 Hz and a second resistive- like range above 1 kHz, as well as a capacitive zone between them. The second resistive-like range is seen 4 Results and discussion as a horizontal line at high frequencies and was firstly 4.1 General trends of models behavior associated to a resistive behavior. However, when the magnitude plot was expressed in dB, see Fig. The transfer functions show that there are three 5(b), the magnitude of the parallel model could be possible behaviors: the capacitive behavior, the appreciated to be continuously decreasing, indicating inductive behavior and the resistive behavior. In a capacitive behavior. Nevertheless, the gain values practical terms, the capacitive behavior is identified are so small that the significance of this behavior by the negative slope while the resistive behavior is could be confusing. This observation warns us to take identified by the zero slope and the inductive behavior advantage of the logarithmic scale representation. is associated to a positive slope. In order to plot the frequency response of the series and parallel models obtained in Section 2, we have to 4.2 Tap water measurements and modelling set the resistance RTP and the capacitance CTP of the system, according to Eq. (6) and Eq. (7). This requires Figs. 4 and 5 also show experimental data for tap the knowledge of Rz and Cz, the resistance and the water. It is observed that data hold a definite resistive capacitance of tap water without bubbles (α = 0). An pattern above 1 kHz of excitation frequency. This first adjustment of both parameters gave us the couple horizontal pattern is well fitted by the series model, of optimized values: Cz = 250µF and Rz = 95Ω, these both in detection point 1 (see Fig. 4) and in detection values were obtained by fitting the experimental data

www.rmiq.org 547 J.C. Rodr´ıguezy A. Soria

J.C. Rodr´ıguezy A. Soria

(a) Linear scale plot. (b) Decibel (dB) scale plot.

Fig. 4: Magnitude plots according to detection point 1, series and parallel models and comparison with experimental values for tap water.

218 ﬀ J.C. Rodr´ıguezydi erent A. fromSoria zero. Below 10 Hz, better adjustment to data should be expected by including a parasitic capacitance (a) Linear scale plot. 219 or a polarization resistance,(b) Decibelas (dB) a scalethird plot. element. While this approach could give a better ﬁtting, it does not contribute 220 to the comparison of the models proposed here and should be considered by further studies. Fig. 4: MagnitudeRodr´ıguez-Sierra plots according and Soria. toRevista detection Mexicana point 1, de seriesIngenier and´ıaQ parallelu ´ımica models Vol. 15, and No. comparison 2 (2016) 543-551 with experimental values for tap water.

212 Nevertheless, as frequency becomes smaller, experimental data in detection point 1 (see Fig. 4) are better fitted 213 by the series model, as compared to the parallel one. Below 10 Hz, experimental data exhibits an inductive behavior 214 and the series model also presents an inductive pattern up to 1 Hz, fitting experimental points, even if fitting is not as 215 accurate as in the resistive range. In this interval the parallel model holds a resistive pattern that cannot fit the data, 216 the adjustment of both models to the experimental data was intent for values of Rz = 95 Ω and Cz = 250 µF. These 217 values were chosen for further studies on the series and parallel transfer functions, even when void fractions are 218 different from zero. Below 10 Hz, better adjustment to data should be expected by including a parasitic capacitance J.C. Rodr´ıguezy A. Soria 219 or a polarization resistance, as a third element. While this approach could give a better fitting, it does not contribute 220 to the comparison of the models proposed here and should be considered by further studies.

(a) Impedance series arrangement, see Fig. 2(a), H (α = 0). (b) Impedance parallel arrangement, see Fig. 2(b), H (α = 0). (a) Linear scale plot. (b) Decibel (dB) scale plot. s p Fig. 5: Magnitude plots according to detection point 2 and comparison with experimental values for tap water. Fig. 4: Magnitude plots according to detection point 1, series and parallel models and comparison with experimental values for tap water.

221 4.3 Void Fraction modelling 212 Nevertheless, as frequency becomes smaller, experimental data in detection point 1 (see Fig. 4) are better fitted 222 The two-phase resistance and capacitance, given by Eqs. (6) and (7), respectively, were substituted into the transfer 213 by the series model, as compared to the parallel one. Below 10 Hz, experimental data exhibits an inductive behavior 223 functions, Eqs. (10) and (11), for detection point 1, as well as in the transfer functions, Eqs. (12) and (13), for 214 and the series model also presents an inductive pattern up to 1 Hz, fitting experimental points, even if fitting is not as 224 detection point 2, for proposed constant parametric values of α = 0.0, 0.1, 0.2, 0.3 . This set of α values embraces 215 accurate as in the resistive range. In this interval the parallel model holds a resistive pattern that cannot fit the data, { } 225 the bubbly flow pattern in bubble columns. Results are shown in Fig. 6(a) for the detection point 1 and in Fig. 6(b) 216 the adjustment of both models to the experimental data was intent for values of Rz = 95 Ω and Cz = 250 µF. These 226 for the detection point 2. 217 values were chosen for further studies on the series and parallel transfer functions, even when void fractions are (a) Impedance series arrangement, see Fig. 2(a), Hs(α = 0). (b) Impedance parallel arrangement, see Fig. 2(b), Hp(α = 0). (a) Linear scale plot. 218 different from zero.(b) Below Decibel 10 (dB)Hz scale, better plot. adjustment to8 data should be expectedManuscrito by including sometido a parasitic a la Revista capacitance Mexicana deIngenier´ıaQu´ımica 219 or a polarizationFig. 5: Magnitude resistance, plots as according a third element. to detection While point this 2 and approach comparison could with give experimental a better fitting, values it does for tap not water. contribute Fig. 4: Magnitude plots according to detection point220 1,to series theFig. comparisonand 4. parallel Magnitude modelsof the plots models and according comparison proposed to detection withhere experimentaland point should beFig. considered 5. Magnitude by plots further according studies. to detection point values for tap water. 1, series and parallel models and comparison with 2 and comparison with experimental values for tap experimental values for tap water. water. 221 4.3 Void Fraction modelling point 2 (see Fig. 5). A horizontal slope is also adjustment to data should be expected by including a 212 Nevertheless, as frequency becomes smaller, experimental data in detection point 1 (see Fig. 4) are better fitted 222 Theshown two-phase by the resistance parallel model and capacitance, in point 1 (Fig. given 4) by but Eqs. (6)parasitic and (7), capacitance respectively, or werea polarization substituted resistance, into the as transfer a 213 by the series model, as compared to the parallel one. Below 10 Hz, experimental data exhibits an inductive behavior 223 functions,its magnitude Eqs. does (10)and not fit (11), the for experimental detection point values. 1, as wellthird as element. in the transfer While this functions, approach Eqs. could (12) give and a better (13), for 214 and the series model also presents an inductive pattern224 up todetection 1InHz the, fitting detection point experimental 2, for point proposed 2 above points, constant 1 evenkHz parametric, if the fitting series isvalues not as of α = 0.0, 0.1, 0.2, 0.3 . This set of α values embraces fitting, it{ does not contribute} to the comparison of the 215 accurate as in the resistive range. In this interval the parallel225 themodel model bubbly fits holds flow experimental pattern a resistive in values bubble pattern (see columns. Fig.that 5),cannot Results while fit thethe are showndata,models in Fig. proposed 6(a) for here the detectionand should point be considered 1 and in Fig. by 6(b) 216 the adjustment of both models to the experimental data226 wasforparallel intent the detection for model values holds point of a2.R negativez = 95 Ω slope,and C evenz = 250 fromµ 10F. Thesefurther studies. 217 values were chosen for further studies on the series and parallelHz, performing transfer as functions, a capacitive even element. when void fractions are Nevertheless, as frequency becomes smaller, 218 different from zero. Below 10 Hz, better adjustment to data8 should be expected byManuscrito including sometido a parasitic a la capacitance Revista Mexicana deIngenier´ıaQu´ımica experimental data in detection point 1 (see Fig. 4) 219 or a polarization resistance, as a third element. While this approach could give a better fitting, it does not contribute are better fitted by the series model, as compared to 4.3 Void Fraction modelling 220 to the comparison of the models proposed here and should bethe considered parallel one. by further Below studies. 10 Hz, experimental data exhibits an inductive behavior and the series model The two-phase resistance and capacitance, given by also presents an inductive pattern up to 1 Hz, fitting Eqs. (6) and (7), respectively, were substituted into experimental(a) Impedance points, series arrangement, even if fitting see Fig.is not 2(a), asH accurates(α = 0). the(b) transfer Impedance functions, parallel arrangement,Eqs. (10) and see (11), Fig. for 2(b), detectionHp(α = 0). as in the resistive range. In this interval the parallel point 1, as well as in the transfer functions, Eqs. (12) Fig.model 5: Magnitude holds a resistive plots pattern according that cannot to detection fit the data, point 2 andand comparison (13), for detection with experimental point 2, for proposed values constantfor tap water. parametric values of α = 0.0,0.1,0.2,0.3 . This set of the adjustment of both models to the experimental data { } α values embraces the bubbly flow pattern in bubble was intent for values of Rz = 95Ω and Cz = 250µF. These values were chosen for further studies on the columns. Results are shown in Fig. 6(a) for the 221 4.3series Void and Fraction parallel transfer modelling functions, even when void detection point 1 and in Fig. 6(b) for the detection fractions are different from zero. Below 10 Hz, better point 2. 222 The two-phase resistance and capacitance, given by Eqs. (6) and (7), respectively, were substituted into the transfer 223 functions, Eqs. (10) and (11), for detection point 1, as well as in the transfer functions, Eqs. (12) and (13), for 548 www.rmiq.org 224 detection point 2, for proposed constant parametric values of α = 0.0, 0.1, 0.2, 0.3 . This set of α values embraces { } 225 the bubbly flow pattern in bubble columns. Results are shown in Fig. 6(a) for the detection point 1 and in Fig. 6(b) 226 for the detection point 2. (a) Impedance series arrangement, see Fig. 2(a), Hs(α = 0). (b) Impedance parallel arrangement, see Fig. 2(b), Hp(α = 0).

Fig. 5: Magnitude plots according to detection point8 2 and comparison withManuscrito experimental sometido values fora la tap Revista water. Mexicana deIngenier´ıaQu´ımica

221 4.3 Void Fraction modelling

222 The two-phase resistance and capacitance, given by Eqs. (6) and (7), respectively, were substituted into the transfer 223 functions, Eqs. (10) and (11), for detection point 1, as well as in the transfer functions, Eqs. (12) and (13), for 224 detection point 2, for proposed constant parametric values of α = 0.0, 0.1, 0.2, 0.3 . This set of α values embraces { } 225 the bubbly ﬂow pattern in bubble columns. Results are shown in Fig. 6(a) for the detection point 1 and in Fig. 6(b) 226 for the detection point 2.

8 Manuscrito sometido a la Revista Mexicana deIngenier´ıaQu´ımica J.C. Rodr´ıguezy A. Soria

Rodr´ıguez-Sierra and Soria. Revista Mexicana deIngenier ´ıaQu ´ımica Vol. 15, No. 2 (2016) 543-551

on the wall of a cylindrical pipe filled with tap water, gave rise to the following conclusions: 1. The series model presents a resistive behavior at excitation frequencies above 1kHz. It also presents an inductive behavior at point 1 for frequencies between 0.1 Hz and 1 Hz, and a capacitive behavior at point 2 between 0.3 Hz and 3 Hz. These facts are consistent with J.C. Rodr´ıguezy A. Soria experimental results for tap water. 2. At point 1 the parallel model presents a resistive behavior at frequencies below 1 Hz and a second (a) Impedance series arrangement, see Fig. 2(a), Hs(α = 0). (b) Impedanceresistive parallel behavior arrangement, above see 10 Fig.Hz. 2(b), ThereHp is(α also= 0). an inductive behavior between these frequencies. Fig. 6: Magnitude plots for series and parallel models for proposed parametric values of α = 0.0, 0.1, 0.2, 0.3 . At point 2 the behavior is resistive{ at frequencies} smaller than 1 Hz and capacitive with very small magnitudes above 1kHz. This behavior was 227 One of the most relevant features to be noticed in both Figs. 6(a)made and 6(b) apparent is that by the using parallel the dB modelscale. is insensitive 228 to changes in volume fraction at frequencies higher than 100 Hz. Thus, the parallel model is only capable to 3. Below 10 Hz an inductive experimental 229 Hz discriminate the void fraction at excitation frequencies below 10 behavior, where reactive is observed behavior at point becomes 1. This relevant behavior and 230 polarization of electrodes is to be considered important. However,follows it is also the in general this range trend of predicted excitation by the frequencies series 231 that the parallel model presents lack of fitting, as shown above, in Figs.model 4 and and 5. cannot Therefore, be fitted the parallel by the model parallel does 232 not fit the experimental behavior at low frequencies and is insensitivemodel. to void Moreover, fraction below at high 1 Hz frequencies.the parallel Both 233 drawbacks are fixed by the series model. model presents a resistive behavior that is not (a) Impedance series arrangement, see Fig. 2(a), Hs(α = 0). (b) Impedance parallel arrangement, see Fig. 2(b), Hp(α = 0). consistent with experimental observations. 4. Above 1kHz a resistive experimental behavior Fig. 6: Magnitude plots for series and parallel models forFig. proposed 6. Magnitude parametric plots values for of seriesα = 0.0 and, 0.1 parallel, 0.2, 0.3 . 234 5 Conclusions { } at point 2 is observed. This behavior is fitted by models for proposed parametric values of α = . , . , . , . the series model and is not fitted by the parallel 235 The comparison0 0 0 1 0 2 0 of3 the. series model and the parallel model for a basic circuit composed just by a resistor and 227 One of the most relevant features to be noticed in both{ Figs. 6(a) and 6(b)} is that the parallel model is insensitive one which hold a capacitive behavior with very 236 a capacitor, as the primary elements of an electrical system arranged by a set of plates flushed on the wall of a 228 to changes in volume fraction at frequencies higher than 100OneHz of. the Thus, most the relevant parallel features model is to only be noticed capable to small magnitude. 237 229 discriminate the void fraction at excitation frequenciescylindrical belowin both 10 pipeHz Figs., wherefilled 6(a) with reactive and tap behavior water, 6(b) is gave becomesthat rise the to relevant parallel the following and conclusions: 5. While the parallel model presents insensitivity 230 polarization of electrodes is to be considered important. However,model is itinsensitive is also in to this changes range of in excitation volume fraction frequencies 238 1. The series model presents a resistive behavior at excitationto frequencies the gas volume above fraction 1 kHz at. higher It also frequencies presents an 231 at frequencies higher than 100 Hz. Thus, the that the parallel model presents lack of fitting, as shown above, in Figs. 4 and 5. Therefore, the parallel model does than 100 Hz, the series model is sensitive to 239 . Hz Hz 232 parallelinductive model behavior is only atcapable point to 1 discriminate for frequencies the void between 0 1 and 1 , and a capacitive behavior at point 2 not fit the experimental behavior at low frequencies and is insensitive to void fraction at high frequencies. Both changes in computed volume fraction. This 240 fractionbetween at 0 excitation.3 Hz and frequencies 3 Hz. These below facts 10 areHz consistent, where with experimental results for tap water. 233 drawbacks are fixed by the series model. is an important point to be accounted for in reactive behavior becomes relevant and polarization of further studies on void fraction measurement 241 2. electrodesAt pointis 1 theto be parallel considered model important. presents However, a resistive it behavior at frequencies below 1 Hz and a second resistive capabilities, since sensor calibrations may 242 isbehavior also in this above range 10 ofHz excitation. There is frequencies also an inductive that the behavior between these frequencies. At point 2 the behavior 234 5 Conclusions introduce theoretical expressions for two-phase 243 parallelis resistive model at presents frequencies lack of smaller fitting, as than shown 1 Hz above,and capacitive with very small magnitudes above 1 kHz. This electrical parameters, i.e. Eqs. (6) and (7), 244 in Figs. 4 and 5. Therefore, the parallel model does not 235 The comparison of the series model and the parallel modelbehavior for a basicwas made circuit apparent composed by using just by the adB resistorscale. and into appropriate electrode modelling, in order fit the experimental behavior at low frequencies and is 236 a capacitor, as the primary elements of an electrical system arranged by a set of plates flushed on the wall of a to account for the gas volume fraction, as we 245 3. insensitiveBelow 10 toHz voidan fraction inductive at experimentalhigh frequencies. behavior Both is observed at point 1. This behavior follows the general 237 cylindrical pipe filled with tap water, gave rise to the following conclusions: suggest here. 246 drawbackstrend predicted are fixed by by the the series series modelmodel. and cannot be fitted by the parallel model. Moreover, below 1 Hz the 238 1. The series model presents a resistive behavior247 at excitationparallel model frequencies presents above a resistive 1 kHz. behavior It also presents that is not an consistent6. The with series experimental model should observations. be extended by 239 inductive behavior at point 1 for frequencies between 0.1 Hz and 1 Hz, and a capacitive behavior at point 2 considering a further element, probably 240 between 0.3 Hz and 3 Hz. These facts are248 consistent4. Conclusions withAbove experimental 1 kHz a resistive results for experimental tap water. behavior at point 2 isa observed. parasitic This capacitance behavior or is fitted a polarization by the series 249 model and is not fitted by the parallel one which hold a capacitiveresistance. behavior Nevertheless, with very small the most magnitude. important 241 2. At point 1 the parallel model presents a resistive behaviorThe comparison at frequencies of the below series 1 modelHz and and a secondthe parallel resistive features of the experimental behavior, both

242 behavior above 10 Hz. There is also an inductive250 behavior5. modelWhile betweenfor the a parallel basic these circuit model frequencies. composed presents At justpoint insensitivity by 2 a the resistor behavior to the gas volumeat low fraction and at at higher highfrequencies, frequencies than are 100 wellHz, 243 and a capacitor, as the primary elements of an represented by the simple series model is resistive at frequencies smaller than 1251Hz and capacitivethe series with model very small is sensitive magnitudes to changes above 1 inkHz computed. This volume fraction. This is an important point to be 244 behavior was made apparent by using the dB scale. electrical system arranged by a set of plates flushed discussed here. 252 accounted for in further studies on void fraction measurement capabilities, since sensor calibrations may 253 introduce theoretical expressions for two-phase electrical parameters, i.e. Eqs. (6) and (7), into appropriate 245 3. Below 10 Hz an inductive experimental behavior is observed at point 1. This behavior follows the generalwww.rmiq.org 549 254 electrode modelling, in order to account for the gas volume fraction, as we suggest here. 246 trend predicted by the series model and cannot be fitted by the parallel model. Moreover, below 1 Hz the 247 parallel model presents a resistive behavior that is not consistent with experimental observations. Manuscrito sometido a la Revista Mexicana deIngenier´ıaQu´ımica 9 248 4. Above 1 kHz a resistive experimental behavior at point 2 is observed. This behavior is fitted by the series 249 model and is not fitted by the parallel one which hold a capacitive behavior with very small magnitude.

250 5. While the parallel model presents insensitivity to the gas volume fraction at higher frequencies than 100 Hz, 251 the series model is sensitive to changes in computed volume fraction. This is an important point to be 252 accounted for in further studies on void fraction measurement capabilities, since sensor calibrations may 253 introduce theoretical expressions for two-phase electrical parameters, i.e. Eqs. (6) and (7), into appropriate 254 electrode modelling, in order to account for the gas volume fraction, as we suggest here.

Manuscrito sometido a la Revista Mexicana deIngenier´ıaQu´ımica 9 Rodr´ıguez-Sierra and Soria. Revista Mexicana deIngenier ´ıaQu ´ımica Vol. 15, No. 2 (2016) 543-551

Acknowledgments Devia, F., Fossa, M. (2003). Design and optimization of impedance probes for void To CONACyT, Mexico, for financial support, Grant fraction measurements. Flow Measurement and CB-2005-C01-50379-Y and Academic Scholarship Instrumentation 14, 139-149. No. 179552 to one of us (JCRS). dos Reis, E., da Silva, C.D. (2014). Experimental study on different configurations of capacitive Nomenclature sensors for measuring the volumetric CTP two phase capacitance concentration in two-phase flows. Flow Cz water capacitance Measurement and Instrumentation 37, 127-134. H transfer function Falcone, G. (2009a). Multiphase Flow j imaginary number unit Fundamentals, In: Developments in Petroleum R resistance Science, (G. Falcone, G.F. Hewitt, C. Alimonti, R two phase resistance TP eds), Elsevier, 54:1-18. Rz water resistance Vin voltage feed Falcone, G. (2009b). Key Multiphase Flow Metering Vk voltage at point k, for k=1, 2 Techniques, In: Developments in Petroleum Z electrical impedance Science. (G. Falcone, G.F. Hewitt, C. Alimonti, Greek symbols eds), Elsevier, 54:47-190. α void fraction ω angular frequency George, D. L., Torczynski, J., Shollenberger, K., O’Hern, T., Ceccio, S. (2001) Three-phase material distribution measurements in a vertical References flow using gamma-densitometry tomography and electrical-impedance tomography. Banasiak, R., Wajman, R., Jaworski, T., Fiderek, International Journal of Multiphase Flow 27, P., Fidos, H., Nowakowski, J., Sankowski, 1903-1930. D. (2014) Study on two-phase flow regime visualization and identification using 3D Jaworek, A., Krupa, A., Trela, M. (2004). electrical capacitance tomography and fuzzy Capacitance sensor for void fraction logic classification. International Journal of measurement in water/steam flows. Flow Multiphase Flow 58, 1-14. Measurement and Instrumentation 15, 317-324.

Bernier, R.J. (1982). Unsteady two-phase flow Ko, M.S., Lee, B.A., Won, W. Y., Lee, Y.G., instrumentation and measurement. Ph.D. Jerng, D.W., Kim, S. (2015). An improved Thesis, Doctor of Philosophy, California electrical-conductance sensor for void-fraction Institute of Technology, Pasadena, California. measurement in a horizontal pipe. Nuclear Engineering and Technology 47, 804-813. ff Cheng H., Hills J.H., Azzopardi B.J. (2002) E ects Kytomaa,¨ H.K., Brennen, C.E.(1988) Some of initial bubble size on flow pattern transition observations of flow patterns and statistical in a 28.9 mm diameter column. International properties of three component flow. Journal Journal of Multiphase Flow 28, 10473-10620. of Fluids Engineering 110, 76-84.

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