{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T21:39:07Z","timestamp":1767908347826,"version":"3.49.0"},"reference-count":25,"publisher":"MDPI AG","issue":"24","license":[{"start":{"date-parts":[[2020,12,9]],"date-time":"2020-12-09T00:00:00Z","timestamp":1607472000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100010198","name":"Ministerio de Econom\u00eda, Industria y Competitividad, Gobierno de Espa\u00f1a","doi-asserted-by":"publisher","award":["ENE2016-79489-C2-1-P"],"award-info":[{"award-number":["ENE2016-79489-C2-1-P"]}],"id":[{"id":"10.13039\/501100010198","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Sensors"],"abstract":"<jats:p>In this paper we perform an analysis of the conductance probes used in two-phase flow applications especially for two-phase flow tomography of annular flow, to measure the waves produced in the interface with different boundary conditions without perturbing the flow, and in addition we examine the holdup applications as measuring the average void fraction in a given region. The method used to obtain the detector conductance between the electrodes is to solve analytically the generalized Laplace equation in 3D with the boundary conditions of the problem, and then to obtain the average potential difference between the detector electrodes. Then, dividing the current intensity circulating between the emitter and the receiver electrodes by the average potential difference yields the probe conductance, which depends on the geometric and physical characteristics of the measured system and the probe. This conductance is then non-dimensionalized by dividing by the conductance of the pipe full of water. In this way a set of analytical expression have been obtained for the conductance of two-plate sensors with different geometries and locations. We have performed an exhaustive comparison of the results obtained using the equations deduced in this paper with the experimental data from several authors in different cases with very good agreement. In some cases when the distribution of bubbles is not homogeneous, we have explored the different alternatives of the effective medium theory (EMT) in terms of the self-consistent EMT and the non-consistent EMT.<\/jats:p>","DOI":"10.3390\/s20247042","type":"journal-article","created":{"date-parts":[[2020,12,9]],"date-time":"2020-12-09T09:17:58Z","timestamp":1607505478000},"page":"7042","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Analysis of Conductance Probes for Two-Phase Flow and Holdup Applications"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4512-7991","authenticated-orcid":false,"given":"Jos\u00e9-Luis","family":"Mu\u00f1oz-Cobo","sequence":"first","affiliation":[{"name":"Instituto de Ingenier\u00eda Energ\u00e9tica, Universitat Polit\u00e9cnica de Valencia, 46022 Valencia, Spain"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7028-9063","authenticated-orcid":false,"given":"Yago","family":"Rivera","sequence":"additional","affiliation":[{"name":"Instituto de Ingenier\u00eda Energ\u00e9tica, Universitat Polit\u00e9cnica de Valencia, 46022 Valencia, Spain"}]},{"given":"Cesar","family":"Berna","sequence":"additional","affiliation":[{"name":"Instituto de Ingenier\u00eda Energ\u00e9tica, Universitat Polit\u00e9cnica de Valencia, 46022 Valencia, Spain"}]},{"given":"Alberto","family":"Escriv\u00e1","sequence":"additional","affiliation":[{"name":"Instituto de Ingenier\u00eda Energ\u00e9tica, Universitat Polit\u00e9cnica de Valencia, 46022 Valencia, Spain"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,9]]},"reference":[{"key":"ref_1","unstructured":"Collier, J.G. (1981). Convective Boiling and Condensation, McGraw-Hill International Book Company. [2nd ed.]."},{"key":"ref_2","unstructured":"Hewitt, G.G., and Hall-Taylor, N.S. (1970). Annular Two-Phase Flow, Pergamon Press."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"112","DOI":"10.1016\/j.nucengdes.2019.03.008","article-title":"Characterization of the gas-liquid interfacial waves in vertical upward co-current annular flows","volume":"346","author":"Cuadros","year":"2019","journal-title":"Nucl. Eng. Des."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"110224","DOI":"10.1016\/j.expthermflusci.2020.110224","article-title":"Experimental study of the effects produced by the changes of the liquid and gas superficial velocities and the surface tension on the interfacial waves and the film thickness in annular concurrent upward vertical flows","volume":"120","author":"Rivera","year":"2021","journal-title":"Exp. Therm. Fluid Sci."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"570","DOI":"10.1016\/j.ijmultiphaseflow.2010.03.004","article-title":"Time spatially resolved measurements of interfacial waves in vertical annular flow","volume":"36","author":"Belt","year":"2010","journal-title":"Int. J. Multiph. Flow"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"351","DOI":"10.1002\/aic.690240228","article-title":"An electroconductivity technique for the measurements of axial variation of holdups in three-phase fluidized beds","volume":"24","author":"Begovich","year":"1978","journal-title":"AIChE J."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1016\/0301-9322(92)90037-H","article-title":"A conductance probe for measuring liquid fraction in pipes and packed beds","volume":"18","author":"Tsochatzidis","year":"1992","journal-title":"Int. J. Multiph. Flow"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1016\/S0955-5986(98)00011-9","article-title":"Design, and performance of a conductance probe for measuring the liquid fraction in two-phase gas-liquid flow","volume":"9","author":"Fossa","year":"1998","journal-title":"Flow Meas. Instrum."},{"key":"ref_9","unstructured":"Lina, Y., and Yingwei, L. (2009, January 26\u201328). On conductance probe measurement model for measuring oil-water annular flow. Proceedings of the ICISE 2009, 1st International Conference on Information Science and Engineering, Nanjing, China."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"903","DOI":"10.1088\/0022-3735\/6\/9\/030","article-title":"The theory and application of conductance probes for the measurement of liquid film thickness in two-phase flow","volume":"6","author":"Coney","year":"1973","journal-title":"J. Phys. E Sci. Instrum."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Lee, K.B., Kim, J.R., Park, G.C., and Cho, K.C. (2017). Feasibility test of a liquid film thickness sensor on a flexible printed circuit board using a three-electrode conductance method. Sensors, 17.","DOI":"10.3390\/s17010042"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"804","DOI":"10.1016\/j.net.2015.06.015","article-title":"A improved Electrical Conductance Sensor for Void Fraction Measurement in a Horizontal Pipe","volume":"47","author":"Ko","year":"2015","journal-title":"Nucl. Eng. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Lee, Y.G., Won, W.Y., Lee, B.A., and Kim, S. (2017). Dual conductance sensor for simultaneous measurement of void fraction and structure velocity of downward two-phase flow in a slightly inclined pipe. Sensors, 17.","DOI":"10.3390\/s17051063"},{"key":"ref_14","unstructured":"Da Silva, M.J. (2008). Impedance Sensors for Fast Multiphase Flow Measurement, and Imaging. [Ph.D. Thesis, Technical University of Dresden]."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2833","DOI":"10.1016\/j.apm.2011.09.069","article-title":"Analytical estimation of liquid film thickness in two-phase annular flow using electrical resistance measurement","volume":"36","author":"Wang","year":"2012","journal-title":"Appl. Math. Model."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"1589","DOI":"10.1088\/0022-3727\/31\/13\/013","article-title":"Theoretical evidence for Lichteneker mixture formulae\u2019 based on the effective medium theory","volume":"31","author":"Zakri","year":"1998","journal-title":"J. Phys. D Appl. Phys."},{"key":"ref_17","unstructured":"Maxwell, J.C. (1882). A Treatise on Electricity and Magnetism, Clarendon Press."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.mser.2008.07.001","article-title":"Predictions of effective physical properties of complex multiphase materials","volume":"63","author":"Wang","year":"2008","journal-title":"Mater. Sci. Eng. R"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1016\/S1359-6462(98)00147-X","article-title":"Electrical conductivity of two-phase composite materials","volume":"39","author":"Kovacik","year":"1998","journal-title":"Scr. Mater."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"636","DOI":"10.1002\/andp.19354160705","article-title":"Calculation of different physical constants of heterogeneous substances","volume":"24","author":"Bruggeman","year":"1935","journal-title":"Ann. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Wang, D., Jin, N., Zhai, L., and Ren, Y. (2020). Salinity Independent Flow Measurement of Vertical Upward Gas-Liquid Flows in a Small Pipe Using Conductance Method. Sensors, 20.","DOI":"10.3390\/s20185263"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1016\/S0955-5986(03)00020-7","article-title":"Void fraction measurement using the impedance method","volume":"14","author":"Yang","year":"2003","journal-title":"Flow Meas. Instrum."},{"key":"ref_23","unstructured":"Wang, D.Y., Jin, N.D., Zhai, L.S., and Ren, Y.Y. (2019, January 3\u20137). Salinity independent flow measurement of vertical upward gas-liquid flows in a small pipe using conductance method. Proceedings of the 11th International. Symposium on Measurement Techniques for Multiphase Flow, Zhenjiang, China."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Yang, Q.Y., Jin, N.D., Zhai, L.S., Ren, Y.Y., Yu, C., and Wei, J.D. (2020). Measurement of Water Velocity in Gas\u2013Water. Two-Phase Flow with the Combination of Electromagnetic Flowmeter and Conductance Sensor. Sensors, 20.","DOI":"10.3390\/s20113122"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"49","DOI":"10.1109\/42.52982","article-title":"Electrical impedance tomography of translationally uniform cylindrical objects with general cross-section boundaries","volume":"9","author":"Ider","year":"1990","journal-title":"IEEE Trans. Med. Imaging"}],"container-title":["Sensors"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/24\/7042\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T10:42:43Z","timestamp":1760179363000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1424-8220\/20\/24\/7042"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,12,9]]},"references-count":25,"journal-issue":{"issue":"24","published-online":{"date-parts":[[2020,12]]}},"alternative-id":["s20247042"],"URL":"https:\/\/doi.org\/10.3390\/s20247042","relation":{},"ISSN":["1424-8220"],"issn-type":[{"value":"1424-8220","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,12,9]]}}}