Experimental validation and critical analysis of inverse method in mass transfer using electrochemical sensor Emna Berrich a , Fethi Aloui a,b,⇑ , Jack Legrand c a LUNAM Université, Université de Nantes, CNRS, GEPEA, UMR 6144, École des Mines de Nantes, 4 rue Alfred KASTLER, BP20722 44307 Nantes Cedex 03, France b Université de Valenciennes et du Hainaut Cambrésis, TEMPO, EA4542, Le Mont Houy, 59313 Valenciennes Cedex 09, France c LUNAM Université, Université de Nantes, CNRS, GEPEA, UMR 6144, CRTT, BP 406 44602 Saint-Nazaire Cedex, France article info Article history: Received 12 August 2011 Received in revised form 16 June 2012 Accepted 1 July 2012 Available online 10 July 2012 Keywords: Wall shear rate Electrochemical method Parallel-plate rheometer Inverse method Mass transfer abstract Rehimi et al. [1] numerically studied the frequency response of an electrochemical probe. They applied an inverse sequential algorithm of the convection diffusion equation to simulated periodic mass transfer sig- nals in order to determine the wall shear rate. This paper presents an experimental validation of the inverse method in mass transfer and a critical analysis of its advantages and limits of application. An elec- trochemical sensor (probe) was used to determine the mass transfer from the current delivered by the probe. The mass transfer is related to the wall shear rate via the convection diffusion equation. The data- base, obtained by electrochemical method also known as ‘‘Electro-Diffusion Method’’, was exploited to validate the inverse method experimentally. The inverse method was checked over a specific range of Péclet numbers varied from 4.58  10 3 to 1.06  10 5 with a well-controlled shear flow with known wall shear rate. An optimized computational algorithm using Matlab Ò was developed for the post-processing, the filtering of the electrochemical results and for programming available models, mostly used in polar- ography for the determination of the wall shear rate (Lévêque [2]; Sobolik et al. [3]; Deslouis et al. [4]). The comparison of these methods with the inverse method and the experimental wall shear rate allows a critical analysis of the restrictions of the different approaches. We demonstrated experimentally, that the difference between the real wall shear rate and the quasi-steady one of Lévêque [2], can reach 9% for dimensionless frequencies f  ¼ fl 2 D 6 205 and oscillation amplitudes b P 0.3. For low frequencies (f / 6 205), Sobolik et al. [3] solution and Deslouis et al. [4] transfer function are in agreement with the experimental wall shear rate. The inverse method is validated for high frequencies of oscillations, for which the linear approaches lead to amplitude attenuation and phase shift of wall shear stress fluctua- tions with respect to the experimental one. Ó 2012 Elsevier Inc. All rights reserved. 1. Introduction The electrochemical method is one of the few non-intrusive techniques used for the measurement of the local wall shear rate, which is a very important parameter for flow characteristics in many experimental devices. The method is based on the determi- nation of the limiting diffusion current delivered by a small probe placed on an inert wall in contact with the liquid flow. By solving the convection–diffusion equation in steady regime and without the axial diffusion term, a solution, the ‘‘Lévêque solution’’ [2], relating the limiting diffusion current and the wall shear rate was proposed for high Péclet numbers and by neglecting the axial diffusion. Sobolik et al. [3] have introduced another technique based on the correction of the wall shear rate obtained by the Lévê- que [2] solution by adding a term deduced from the transitory response: S Sob ðtÞ¼ S q ðtÞþ 2 3 hðtÞ @S q ðtÞ @t   ð1Þ where S q ¼ S Lev ¼ D l 2 ShðtÞ 0:807   3 ð2Þ and hðtÞ¼ 0:486l 2 3 D  1 3 S q ðtÞ  2 3 ð3Þ This method correctly predicts the wall shear rate at high aver- age Péclet numbers when the sampling rate is sufficient. The most common approaches related to the wall shear stress probes are focused on the research of transfer functions between 0894-1777/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.expthermflusci.2012.07.001 ⇑ Corresponding author at: Université de Valenciennes et du Hainaut Cambrésis, TEMPO, EA4542, DF2T, Le Mont Houy, 59313 Valenciennes Cedex 09, France. Tel.: +33 3 27 51 19 62; fax: +33 3 27 51 19 61. E-mail address: Fethi.aloui@univ-valenciennes.fr (F. Aloui). Experimental Thermal and Fluid Science 44 (2013) 253–263 Contents lists available at SciVerse ScienceDirect Experimental Thermal and Fluid Science journal homepage: www.elsevier.com/locate/etfs