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Calibration of electrodiffusion probes for turbulent flow measurements
Flow Measurement and Instrumentation 35 (2014) 76–83
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Calibration of electrodiffusion probes for turbulent flow measurements A. Dib a,b,n, S. Martemianov b,nn, L. Makhloufi a, B. Saidani a Laboratoire d0 Electrochimie, Corrosion et de Valorisation Energétique (L.E.C.V.E), Département de Génie des Procédés, Faculté de la Technologie, Université A. Mira Bejaia, Algérie b Institut P0 , UPR 3346 CNR S, Université de Poitiers, ENSMA,- Branche FluideBâtiment Mécanique, 3ème étage, Campus Sud, du Recteur Pineau, 86022 Poitiers, France a
art ic l e i nf o
a b s t r a c t
Article history: Received 20 May 2012 Received in revised form 4 September 2013 Accepted 21 November 2013 Available online 27 December 2013
Calibration of electrodiffusion (ED) probes with respect to the wall velocity gradient measurements has been experimentally studied in a fully developed turbulent channel flow over the Reynolds number range 14000–23000. In steady state conditions, direct calibration concerning the mean wall velocity gradient can be provided using simultaneous transient and steady state diffusion limiting current measurements. Indirect calibration of the ED probes regarding the turbulent fluctuation has been undertaken using a spectral analysis performed with probes of different size and geometry. This indirect calibration method has been supported using the measurements of thermo-dependence of molecular diffusivity. In this case, only one ED probe can be used for calibration. Dynamic calibration provides necessary information about critical longitudinal dimension of ED probes. When the longitudinal dimension of the probes exceeds the critical length, the probes becomes sensitive to the normal fluctuation as well. This fact has been confirmed using different orientation of the rectangular ED probe with respect to the flow direction. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Electrodiffusion Calibration Wall velocity gradient Near wall turbulence
1. Introduction Electrodiffusion (ED) technique is well adapted for near wall hydrodynamics and mass transfer measurements [1–3]. The possible applications concern chemical engineering, drag reduction by means of surfactants, bioengineering, micro-fluidics, multiphase flows, corrosion and deposition processes [4–11]. Moreover, this method is used as an advanced tool providing unique experimental information for verification and adjustment of theoretical models and computational codes related to near wall turbulence and mass transfer [12–15]. Different methods for ED measurements have been developed in non-stationary flow conditions [16–18]. For turbulent flows, the transfer function related the power spectral density (PSD) of wall velocity gradient and current fluctuations is usually used [2,16,19–22]. Nevertheless, this approach meets the difficulties linked to the unwanted changes of experimental conditions (temperature and depolarizer concentration) and reliability of the probes due to the alteration of their surface and geometry caused by gloating, polishing,
n Corresponding author at: Laboratoire d0 Electrochimie, Corrosion et de Valorisation Energétique (L.E.C.V.E), Département de Génie des Procédés, Faculté de la Technologie, Université A. Mira Bejaia, Algérie. Tel.: þ213 34 21 57 04; fax: þ213 34 20 51 93. nn Principal corresponding author. Tel.: þ 33 5 49 45 39 04; fax: þ33 5 49 45 35 39. E-mail addresses: dib_hafi[email protected] (A. Dib), [email protected] (S. Martemianov).
0955-5986/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2013.11.008
bad implantation and aging. Thus, the calibration of ED probes is necessary. Calibration of ED probes with respect to time average data can be provided directly using the Lévêque method and additional control of the calibration characteristics, such as active surface area, can be achieved using a voltage step transient method [23,24]. However, the significant difference between active surface area obtained from Leveque and Cottrell methods meet the ambiguity related to the transfer function which requires knowledge of the probe dimension. It should be noted that the direct calibration of ED probes regarding to the turbulent fluctuation measurements is not possible due to inexistence of the etalon; then the problem is still open. In this paper the calibration of ED probe0 s for measuring spectra of dynamic fluctuations in turbulent flow is proposed by using probes of different sizes and geometry. For this purpose, a spectral analysis is performed to check its reliability. The proposed method is supported by taking in consideration the thermo dependence of ED measurements. This is an attractive way as only one ED probe can be used for calibration in this case.
2. Electrodiffusion method The ED method consists in measuring of limiting diffusion current provided by fast electrochemical reaction taking place at the surface of small probes flush mounted at the wall. The reduction of ferricyanide ions on a platinum cathode is the most popular
A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
Nomenclature probe active surface (m2) depolarizer concentration (mol/m3) diffusion coefficient of depolarizer (m2/s) hydraulic diameter of channel (m) diameter of circular probe (m) voltage (V) Faraday number, F ¼ 96485 (C/mol) frequency (Hz) transfer function limiting diffusion current (A) shape factor length of rectangular probe (m) width of rectangular probe (m) number of electrons involved in electrochemical reaction wall velocity gradient (s 1) temperature (K) time (s) mean flow velocity (m/s) hydraulic Reynolds number ð ¼ UDh =νÞ power spectra density of current fluctuations (A2/Hz) power spectra of wall velocity gradient fluctuations (Hz)
A c D Dh d E F f H IL k L l n S T t U Reh Wii Wss
Greek symbols
λ ρ
skin friction coefficient fluid flow density (kg/m3)
electrochemical reaction used for such flow diagnostics [1]: FeðCNÞ6 3 þ e ⇀FeðCNÞ6 4 :
I L ¼ ςS
;
ς ¼ knFcb D
2=3
ð1Þ
w
1=3
ð2Þ
where w ¼d, A ¼ π ðd=2Þ , k¼ 0.686 for a circular ED probe; w¼l, A¼ lL, k ¼0.807 for a rectangular ED probe in perpendicular position and w¼L, A ¼lL, k¼0.807 for a rectangular ED probe in parallel position. If the velocity gradient is time dependent, the Leveque solution Eq. (2) generally cannot be applied to deduce S(t) from I(t). In particular, this is the case of high frequency fluctuations when the concentration boundary layer acts as a filter and damps the current fluctuations. Thus, the current can be defined as a convolution product between the time-dependent velocity gradient and the impulse response of the system [16,22]. The impulse response of the system depends on the probe0 s geometry, mean wall velocity gradient and physico-chemical properties of the electrolyte solution. This paper is limited to the diffusion boundary layer response in the frequency domain where the power spectra1 density of the current fluctuations, Wii, is connected to that of the velocity gradient fluctuations, Wss, by 2
W ii ¼ jHðf Þj2 W ss ;
cinematic viscosity (m2/s) Lévêque coefficient (A s1/3) Cottrell coefficient (A s1/2) distance in the pressure drop (m) pressure drop (Pa) dimensionless frequency of circular probe dimensionless frequency of rectangular probe
Subscripts Lev Cot geo b h
Lévêque Cottrell geometric bulk hydraulic
Superscripts 4
dimensionless units mean value
Mathematical symbols and abbreviations R? ci R PSD ED
rectangular probe in perpendicular direction circular probe number i rectangular probe in parallel direction // power spectral density electrodiffusion probes
assuming that the axial molecular diffusion is negligible:
The mean limiting current I L is related to the mean velocity gradient S by the well-known Lévêque formula [25]: 1=3
ν ς β ΔL ΔP s s0
77
ð3Þ
where jHðf Þj is the amplitude of the transfer function of the diffusional boundary layer. This transfer function is obtained [16,22,26] by the resolution of the convective diffusion equation for the concentration fluctuations, after its linearization and by
∂C 0 ∂C 0 ∂2 C 0 ∂C þ Sx y D 2 ¼ s0x y ∂x ∂t ∂x ∂y 0 0 ∂C 1 ∂sx ∂sz 2 ∂C ; þ y s0z y ∂z 2 ∂x ∂z ∂y
ð4Þ
where s0x and s0z are the amplitude of the velocity gradient fluctuations in the axial, x, and the transversal, z, directions, and C being the mean concentration. The resolution of Eq. (4) depends on the probe0 s geometry. When the ED probe is segmented [21,27], both components of the velocity fluctuations are important and can be measured. The last term on the right-hand side of Eq. (2) deals with the normal velocity fluctuations and is usually neglected. Nevertheless, for relatively long ED probes the influence of the normal velocity fluctuations should be taken into account [22,28]. The difficulty of this approach is that the linearization assumption aforementioned is not always valid, in particular when the amplitude of the fluctuations becomes important [18,29–31]. Therefore, development of the frequency response correction using an inverse methods [18,32–37] or Sobolik0 s solution [17,38,39] leads to well resolved the issue of the dynamic behavior of the ED probe and allows correctly prediction of the wall velocity gradient with fine wall statistics analysis. In the present paper, the use of simple small ED probes (circular or rectangular) makes it possible to only preserve, in the convective diffusion equation , the longitudinal wall velocity gradient with the diffusion term in the normal direction to the flow [40,41]. In these conditions, the current spectra is related only with longitudinal velocity gradient fluctuations through Eq. (3). Moreover, most of the measured fluctuation intensities of the
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diffusional current were below 12%, which means that the linearity conditions are fulfilled and that the measurements could be treated in the framework of a linear theory [29,42]. So, the linear distortion of the instantaneous velocity gradient signal, due to the mass transfer inertia of the diffusional layer, can be then examined in terms of the aforementioned transfer function [20,43]. The prediction of the later in a wide frequency range was obtained by a numerical integration of Eq. (4) [16,22,26]. The amplitude of the transfer function jHðf Þj is expressed as follows:
For rectangular ED probe: Hðs0 Þ ¼ ð1 þ 0:056s02 þ 0:0021s04 Þ 1=2 ; s0 o 6 Hð0Þ
Hðs0 Þ Hð0Þ ¼
s ¼ 2π f 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffi ð3:99 2Þ2 þ ð3:99 2 7:43 s0 Þ2 ; 2ðs0 Þ3=2 w2 DS
s0 Z 6
ð7Þ
2
Hð0Þ
s0 ¼ 2π f
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffi ð5:3 2Þ2 þ ð5:3 2 8:832 sÞ2 ; 2ðsÞ3=2 d
2
DS
2
ð6Þ
!
For circular ED probe: HðsÞ ¼ ð1 þ 0:049s2 þ 0:0006s4 Þ 1=2 ; s o 6 HðsÞ Hð0Þ ¼
ð5Þ
sZ6
ð8Þ
ð9Þ
! ð10Þ
here s0 and s are respectively dimensionless frequency of rectangular and circular probes; jHð0Þj ¼ I L =3S being the amplitude of the transfer function for the quasi-steady state conditions.
3. Experimental details The test bench is a recirculation flow loop (see Fig. 1) consisting of 460 mm length Plexiglas rectangular channel with a crosssection 15.3 mm 4 mm (aspect ratio is equal to 4). The channel is formed by a superposition of three parts. The lower part (1) contains two points for pressure drop measurements distant of 150 mm and centered on the channel axis. The higher part (2) of the channel has 20 mm diameter hole which is used for insertion of the ED probes (3). The higher and the lower parts are separated by the rectangular plate (4) giving place to a cross-section of the channel. The temperature in each experiment have been adjusted by means of a thermostat (Ecoline R312) allowing regulation of electrolyte temperature in the tank with the volume of V tank ¼ 10 l. The additional control of the temperature have been provided using a Curiolis flow-meter (ELITE, CMF050 model) introduced in hydraulic circuit outside the tank. The liquid have been pumped from the tank to a channel test section which have very small volume V channel compared to the tank volume ðV channel ¼ 0:003V tank Þ. The duration of each experiment does not exceeded 30 s, during this time the temperature variations recorded by Curiolis flow-meter have been less than 0.1 1C and the temperature in the channel has been the same as in the tank. Further details regarding the hydraulic calibration of the rectangular channel may be found in [24]. 3.1. Electrochemical probes The nomenclature of rectangular and circular ED probes fabricated from platinum sheet and wire is given in Table 1. The platinum probe is insulated by a polymeric film and glued by Epoxy resin into a large stainless steel tube using as an auxiliary electrode with the surface about 4 10 4 m2. A meticulous and progressive polishing of probes is carried out using abrasive papers with size of decreasing grains. The ED probes are flush mounted with the upper wall of the channel. The geometrical
Fig. 1. Experimental set-up.
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Table 1 Active surface area of ED probes obtained using optic, Cottrell and Lévêque method. Probes
Ageo 106 ðm2 Þ
(β 8 0.35) 106 ðA s1=2 Þ
(ς8 0.35) 106 ðA s1=3 Þ
ACot 106 ðm2 Þ
ALev 106 ðm2 Þ
c1 c2 c3 R?
0.53 0.19 0.04
26.50 9.21 2.20
9.25 3.52 0.90 3.76
0.67 0.23 0.06
0.46 0.14 0.03
0.12
0.13 0.85 ¼ 0.11
0.11 0.99 ¼ 0.11
4.60
R==
2.01
Fig. 2. Photographs of circular and rectangular ED probes.
surface of ED probes is determined by a numerical image treatment of optical microscope photographs and the rectangular ED probe can be placed either on the direction parallel or perpendicular to the mean flow (see Fig. 2). 3.2. Electrochemical system The liquid used for electrochemical flow measurements is a dilute solution containing 25 mol/m3 equimolar potassium ferro/ ferricyanide and 300 mol/m3 potassium sulphate as supporting electrolyte. The viscosity of the solution has been measured with an Ubbelholde capillary viscosimeter and the following linear equation is obtained for the temperature range T A (280–320) K with 93% accuracy [24]: T T0 νðTÞ ¼ νðT 0 Þ þ ξ1 ; ð11Þ T0
Fig. 3. Time-evolution of transient limiting diffusion current for circular and rectangular ED probes, T¼25 1C.
The experimental procedure starts with recording of Cottrell curves in order to check the state of ED probes. This method is based on the application of a voltage step (which corresponds to the diffusion plateau) and the recording of the current time evolution [46]. For very short times, the transient current can be calculated using the well known Cottrell asymptote: rffiffiffiffi D I L ðtÞ ¼ β t 1=2 ; β ¼ nFAcb : ð13Þ
where ξ1 ¼5 10 6 m2/s1 and the reference viscosity at T0 ¼298 K is equal to 10 6 m2/s. The temperature dependence of the molecular diffusion coefficient of depolarizer D in the temperature range T A (280–320) K has been obtained using the classical rotation disc electrode method [44,45]. The obtained results fit with 96% accuracy the following equation [24]: T T 0 DðTÞ ¼ DðT 0 Þ þ ξ2 ; ð12Þ T0
4. Electrodiffusion probes calibration
where ξ2 ¼4.9 10 and the reference viscosity at T0 ¼298 K is equal to 8.33 10 6 m2/s.
4.1. Calibration with respect to mean velocity gradient measurements
9
3.3. Data acquisition and signal treatment Data acquisition of the instantaneous limiting diffusion current delivered by the ED probe has been provided at the sample frequency equal to 103 Hz using a Solartron SI 1480 MultiStat potentiostat controlled by Corrwarre software. A constant polarization voltage (E¼ 0.6 V) is imposed in all experiments in order to ensure the limiting diffusion regime. Once the limiting diffusion current is recorded, the mean values of the wall velocity gradient can be determined by means of Eq. (2). Then, the current fluctuations can be treated and its power spectral density Wii can be calculated. The frequency response of the ED probes is taken into account to restore the spectra of the wall velocity gradient fluctuations Wss using the transfer function given by Eq. (3).
π
In this section we discuss the procedure of the calibration of the ED probes with respect to the mean wall velocity gradient measurements. The results of voltage step measurements are presented on Fig. 3; the good agreement of the experiment with Cottrell law is justified for all probes. Some deviation from the Cottrell behavior observed for the very short times can be explained by the double layer effects. These results justify the reliability of the sensor fabrication and can be used for the determination of the active surface area A. Our installation allows determining the mean wall velocity gradient, in the section of electrochemical measurements, by measuring of the pressure drop ΔP : S ¼ Dh
ΔP ; 4ρνΔL
ð14Þ
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therefore, it is possible to provide the direct calibration of the ED probes with respect to mean wall velocity gradient measurements using Lévêque law; the corresponding results are presented in Fig. 4. For the rectangular probe Lévêque measurements have been provided for the both positions: probe is placed parallel or perpendicular to the mean flow direction. It can be noted that all the probes satisfy correctly Lévêque law. It should be noted that the thermo-dependence of the ED measurements put on the question of the temperature effects compensation in particular for the measurements in nonisothermal conditions. Moreover, possible time variations of the active surface A and the bulk concentration cb should be taken into account. The conjoint use of Lévêque, Cottrell and rotating disc electrode measurements allows solving this problem [24]. The experimental confirmation of Lévêque and Cottrell laws give the definitive justification that the ED probes provide reliable measurements of the mean velocity gradient. Unusual transient behavior (the current IL(t) does not follow t 1=2 law) can be used as the indicator of the sensor defect [23]. Although the steady state calibration is successfully accomplished, the determination of the active surface area remains a delicate problem. The results of the
Fig. 4. Lévêque curves for circular and rectangular ED probes at T¼ 25 1C.
probe dimensions measurements obtained by means of Cottrell, Lévêque and optical methods are presented in Table 1; the significant difference for the probe surface area should be noted with respect to using method of determination. 4.2. Calibration with respect to measurements of wall velocity gradient fluctuations 4.2.1. Influence of probe geometry It is well known that the current fluctuations give an access to the turbulent fluctuations of the wall velocity gradient [1,3]. Nevertheless, in many cases, the concentration boundary layer inertia cannot be neglected and should be taken into account. Consequently, different methods have been developed and all meet the problem of sensors calibration with respect to fluctuations measurements. The common approach for the treatment of turbulent fluctuations is based on the using of the transfer function [16,22] for determination of the power spectral density (PSD), it is intimately related to knowing of the sensor geometry and the molecular diffusion coefficient of depolarizer. As we have noted above, unwanted changes in experimental conditions can occur suddenly and the active surface area may be determined with some uncertainty. So, justification of the ED method reliability regarding the turbulent fluctuation measurements is needed. For the spectra analysis of the wall velocity gradient fluctuations this justification can be achieved by using probes with different size and geometry. This justification can be achieved by simultaneous recording the wall velocity gradient fluctuations using probes with different size and geometry. Fig. 5(a) presents, as example, PSD measurements of current fluctuations obtained for Reh ¼ 14 103 using four ED probes. All the results presented in this paragraph concern the measurements provided at the fixed temperature T ¼25 1C. Obviously, the current fluctuations are very sensitive to the probes size, as well as the transfer function which links the current to the wall velocity gradient fluctuations (see Fig. 5(b)). For each probe the transfer function has been calculated using the probes dimensions obtained by means of different methods (Cottrell, Lévêque and optic). It can be noted that the dispersion of the results with respect to this uncertainty is not important (about 9%) and lies within the precision of the PSD recording. Fig. 6(a, b) presents, for two Reynolds numbers, PSD measurements of wall velocity gradient fluctuations obtained by ED probes
Fig. 5. PSD of current fluctuations (a) and transfer function (b) T ¼ 25 1C, Reh ¼ 14 103.
A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
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Fig. 6. Dimensionless PSD of wall velocity gradient fluctuations at T ¼25 1C. (a) Reh ¼ 14 103. (b) Reh ¼ 23 103.
Fig. 7. Thermo-dependence of ED measurements obtained with circular ED probe c1 at Reh ¼14 103. (a) PSD of current fluctuations. (b) Transfer function.
with different size and geometry. The following dimensionless form of PSD is used: W ss4 ¼
Wssðf Þ : W ss ðf -0Þ
ð15Þ
The spectra of the wall velocity gradient fluctuations coincides for all the probes, this results has been confirmed for all tested Reynolds numbers. Coinciding of the PSD curves gives the justification of the reliability of the using ED probes for turbulent fluctuations measurements. This is important, because today the direct calibration of the ED probes with respect to fluctuation measurements is not possible because of the lack of a reference etalon. 4.2.2. Influence of temperature The possibility of the indirect calibration of ED probes with respect to fluctuation measurements can be supported using the thermal effects. Indeed, the transfer function depends on the
molecular diffusivity and thus on the solution temperature. From the other hand, it is reasonable to suppose that the dimensionless PSD of wall velocity gradient fluctuations via Eq. (15) is independent or at least weakly dependent on the temperature. Thus, an additional possibility for checking the reliability of ED probes regarding fluctuation measurements is possible. This possibility consists in varying of the solution temperature and recording PSD fluctuations; it is an attractive way since only one sensor can be used. As an example, Fig. 7(a) presents PSD measurements of current fluctuations recorded at different temperatures using circular ED probe C1. The temperature effects are clearly pronounced for the current fluctuation measurements. The transfer functions have been calculated taking into account the thermo dependence of the depolarizer diffusivity (Fig. 7b), and then the corresponding PSD of the wall velocity gradient fluctuations have been recovered via Eq. (3). The dimensionless PSD of the wall velocity gradient recorded at different temperatures are presented in Fig. 8(a, b). Coinciding of
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the experimental curves gives the additional justification of the reliability of using ED probes for turbulent fluctuations measurements.
4.2.3. Limitations related with probe geometry It is necessary to make a distinction between the unreliability of ED probes and the interpretation of the obtained experimental data. The rectangular ED probes have proved their reliability with respect to the mean velocity gradient measurements in both positions (parallel and perpendicular) with respect to the mean flow. Fig. 9(a, b) presents, for two Reynolds numbers, the results of PSD measurements of the wall velocity gradient fluctuations obtained with the rectangular probe placed in parallel (parallel position) to the mean flow velocity position. In the same figure the PSD measurements previously obtained by circular and rectangular (perpendicular position) ED probes are presented. The significant difference of the
recorded PSD curves with respect to the probe orientation can be noted. Thus, the calibration with respect to the mean measurements is not sufficient for the fluctuation measurements. The mean current of ED probe can be sensitive to the mean flow velocity and, in the same time, the current fluctuations can be related to another hydrodynamic characteristic. The change of the probe orientation manifests itself in the increasing of the longitudinal probe size. Regarding the mean wall velocity gradient measurements the probe satisfies the Lévêque condition, but the current fluctuations do not reflect the fluctuations of the wall velocity gradient. This effect can be interpreted using the results of the theory of the ED probes which takes into account the influence of the normal velocity fluctuations [28]. According to this theory, there is a critical longitudinal length for the validity of the classical transfer function approach. This length is proportional to the viscous sublayer thickness and to the ratio between the intensity of the normal and the longitudinal velocity fluctuations
Fig. 8. Dimensionless PSD of wall velocity gradient fluctuations at different temperatures. The measurements with circular ED probe c1: (a) Reh ¼ 14 103. (b) Reh ¼ 23 103.
Fig. 9. Effect of ED probe orientation on the dimensionless PSD of wall velocity gradient fluctuations at T ¼ 25 1C. (a) Reh ¼14 103. (b) Reh ¼23 103.
A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
on the external border of the viscous sublayer. Once the longitudinal probe length is larger in comparison to this critical length, the current fluctuations is not sensitive to the wall velocity gradient fluctuations but to the normal velocity fluctuations. So, the ED probe can be sensitive to the mean velocity gradient (Lévêque law) but the current fluctuations of this probe cannot be interpreted in terms of the wall velocity gradient fluctuations. 5. Conclusion The transient voltage step method together with the steady state limiting diffusion current measurements can be used successfully for ED probes calibration with respect to mean wall velocity gradient measurements. Reliability of ED turbulent fluctuation measurements can be proved by using probes with different size and geometry. Coinciding of all wall velocity gradient PSD, within the experimental accuracy range, justifies the probe0 s reliability. The temperature effects open additional possibility for checking of the ED probes reliability. Indeed, dimensionless PSD of wall velocity gradient fluctuations is quasi independent on the temperature. Once the reliability of the ED probes is proved, a fine interpretation of PSD measurements becomes possible. In particular, verification of the Lévêque law is not sufficient for the conclusion that the current fluctuations reflect the wall velocity gradient fluctuations. This fact has been clearly confirmed using different orientation of the rectangular ED probe with respect to the flow direction. Changing the probe0 s orientation make it sensitive to normal velocity fluctuations and, thus, open additional possibilities in ED flow diagnostic. Acknowledgments This work pertains to the French Government program “Investissements d0 Avenir” (LABEX INTERACTIFS, reference ANR-11LABX-0017-01). References [1] Mitchell J, Hanratty T. A study of turbulence at a wall using an electrochemical wall shear-stress meter. J Fluid Mech 1966;26:199–221. [2] Fortuna G, Hanratty T. Frequency response of the boundary layer on wall transfer probes. Int J Heat Mass Transf 1971;14:1499–507. [3] Hanratty TJ, Campbell JA. Measurement of wall shear stress, fluid mechanics measurements. Washington: R.J. Goldstein, Hemisphere; 1983. [4] Legrand J, Legentilhomme P, Aouabed H, Ould-Rouis M, Nouar C, Salem A. Electrodiffusional determination of momentum transfer in annular flows: axial developing and swirling decaying flows. J Appl Electrochem 1991;21:1063–7. [5] Adolphe X, Danaila L, Martemianov S. On the small-scale statistics of turbulent mixing in electrochemical systems. J Electroanal Chem 2007;600:119–30. [6] Alekseenko SV, Markovich DM, Evseev AR, Bobylev AV, Tarasov BV, Karsten VM. Experimental study of liquid distribution in a column with a structured packing. Theor Found Chem Eng 2007;41(4):417–23. [7] Boutoudj M, Ouibrahim A, Barbeu F, Deslouis C, Martemianov S. Local shear stress measurements with microelectrodes in turbulent flow of drag reducing surfactant solutions. Chem Eng Process: Process Intensif 2008;47:793–8. [8] Blel W, Le Gentil-Lelièvre C, Bénézech T, Legrand J, Legentilhomme P. Application of turbulent pulsating flows to the bacterial removal during a cleaning in place procedure. Part 1: experimental analysis of wall shear stress in a cylindrical pipe. J Food Eng 2009;90:422–32. [9] Huchet F, Comiti J, Legentilhomme P, Solliec C, Legrand J, Montillet A. Multiscale analysis of hydrodynamics inside a network of crossing minichannels using electrodiffusion method and PIV measurements. Int J Heat Fluid Flow 2008;29:1411–21. [10] Kashinskii O, Kaipova E, Kurdyumov A. Application of the electrochemical method for measuring the fluid velocity in a two-phase bubble flow. J Eng Phys Thermophys 2003;76(6):1215–20. [11] Barbier F, Alemany A, Martemianov S. On the influence of a high magnetic field on the corrosion and deposition processes in the liquid Pb-17Li alloy. Fusion Eng Des 1998;43:199–208. [12] Parys H, Tourwé E, Breugelmans T, Depauw M, Deconinck J, Hubin A. Modeling of mass and charge transfer in an inverted rotating disk electrode (IRDE) reactor. J Electroanal Chem 2008;622:44–50.
83
[13] Legrand J, Martemyanov SA. Turbulent mass transfer in small radius ratio annuli. J Appl Electrochem 1994;24(8):737–44. [14] Martemianov S, Okulov V. Mass transfer ambiguities in swirling pipe flows. J Appl Electrochem 2002;32(1):25–34. [15] Pallares J, Grau F. Frequency response of an electrochemical probe to the wall shear stress fluctuations of a turbulent channel flow. Int J Heat Mass Transf 2008;51:4753–8. [16] Deslouis C, Gil O, Tribollet B. Frequency response of electrochemical sensors to hydrodynamics fluctuations. J Fluid Mech 1990;215:85–100. [17] Sobolik V, Wein O, Cermak J. Simultaneous measurement of film thickness and wall shear stress in wavy flow of non-Newtonian liquids. Collect Czechoslov Chem Commun 1987;52:913–28. [18] Rehimi F, Aloui F, BenNasrallah S, Doubliez L, Legrand J. Inverse method for electrodiffusional diagnostics of flows. Int J Heat Mass Transf 2006;49:1242–54. [19] Ambari A, Deslouis C, Tribollet B. Frequency response of the mass transfer rate in a modulated flow at electrochemical probes. Int J Heat Mass Transf 1986;29:35–45. [20] Nakoryakov VE, Kashinsky N, Kozmenko BK. Electrochemical method for measuring turbulent characteristics of gas-liquid flows, measuring techniques in gas–liquid two phase flows. In: IUTAM symposium, Nancy, France; 1983. p. 695–721. [21] Deslouis C, Huet F, Gil O, Tribollet B. Spectral analysis of wall turbulence with a bicircular electrochemical probe. Exp Fluids 1993;16:97–104. [22] Skurygin E, Martemianov S, Vorontyntsev M, Grafov B. Computing the frequency characteristics of turbulent diffusion layers at microelectrodes. Sov Electrochem 1989;25:685–8. [23] Sobolik V, Tihon J, Wein O, Witcherle K. Calibration of electrodiffusion friction probes using a voltage-step transient. J Appl Electrochem 1998;28:329–35. [24] Dib A, Martemianov S, Makhloufi L, Saidani B. Thermo-compensation of electrodiffusional wall velocity gradient measurements. Int J Heat Fluid Flow 2008;29:219–28. [25] Lévêque M. Transmission de chaleur par convection. Ann des Mines 1928;13:381–412. [26] Nakoryakov VE, Burdukov AP, Kashinsky ON, Geshev PI. Electrodiffusion method of investigation into the local structure of turbulent flows. Novosibirsk: V.G. Gasenko; 1986. [27] Sobolik V, Wein O. Three-segment electrodiffusion probes for measuring velocity fields close to a wall. Exp Fluids 1990;9:43–8. [28] Martemyanov SA, Petrovskiy NV, Grafov BM. Turbulent pulsations of the microelectrode limiting diffusion current. J Appl Electrochem 1991;21 (12):1099–102. [29] Kaiping P. Unsteady forced convective heat transfer from a hot film in nonreversing and reversing shear flow. Int J Heat Mass Transf 1983;26:545–57. [30] Mao Z, Hanratty TJ. Analysis of wall shear stress probes in large amplitude unsteady flows. Int J Heat Mass Transf 1991;34(1):281–90. [31] Mao ZX, Hanratty TJ. The use of scalar transport probes to measure wall shear stress in a flow with imposed oscillations. Exp Fluids 1985;3:129–35. [32] Mao Z, Hanratty TJ. Application of an inverse mass transfer method to the measurement of turbulent fluctuations in the velocity gradient at the wall. Exp Fluids 1991;11:65–73. [33] Maquinghen T. Métrologie tridimensionnelle instationnaire à l0 aide de la méthode polarographique [Ph.D. thesis]. Thèse de doctorat, Université de Valenciennes; 1999. [34] Berrich E, Aloui F, Legrand J. Experimental validation and critical analysis of inverse method in mass transfer using electrochemical sensor. Exp Therm Fluid Sci 2013;44(0):253–63. [35] Berrich E, Aloui F, Legrand J. Inverse method for the dynamical analysis of wall shear rates using three-segment probes in parallel plate rheometer. Chem Eng Sci 2011;66(17):3969–78. [36] Zidouh H, Labraga L, William-Louis M. Unsteady wall shear stress in transient flow using electrochemical method. J Fluids Eng 2009;131(5):1 http://dx.doi. org/10.1115/1.3112387. [37] Keirsbulck L, Labraga L, Gad-el Hak. Statistical properties of wall shear stress fluctuations in turbulent channel flows. Int J Heat Fluid Flow 2012;37(0):1–8. [38] Huchet F, Legentilhomme P, Legrand J, Montillet A, Comiti J. Unsteady flows in milli- and microsystems: analysis of wall shear rate fluctuations. Exp Fluids 2011;51(3):597–610. [39] El Hassan M, Assoum H, Sobolik V, Vétel J, Abed-Meraim K, Garon A, et al. Experimental investigation of the wall shear stress and the vortex dynamics in a circular impinging jet. Exp Fluids 2012;52:1475–89. [40] Ling S. Heat transfer from a small spanwise on an insulated boundary. J Heat Transfer Trans ASME 1963:230–6. [41] Mollet L, Dumargue PMD, Bodiot D. Calcul du flux limite de diffusion sur une microelectrode de section circulaire. Equivalence avec une électrode de section rectangulaire. vérification expérimentale dans le cas du disque toumant en régime laminare. Efectrochim Acta 1974;19:841–4. [42] Rode S, Midoux N, Latifi M, Storck A, Saatdjian E. Hydrodynamics of liquid flow in packed beds: an experimental study using electrochemical shear rate sensors. Chem Eng Sci 1994;49(6). [43] Deslouis C, Gil O, Tribollet B. Frequency response of electrochemical sensors in a cone-and-plate modulated flow. Int J Heat Mass Transf 1990;33(11):2525–32. [44] Coeuret F, Storck A. Eléments de génie électrochimique. Paris: Tec et Doc Lavoisiers; 1984. [45] Levich V. Physicochemical hydrodynamics. Englewood, USA: Prentice-hall; 1962. [46] Cottrell F. Courant de réponse à une polarization galvanique traité comme un problème de diffusion. Z Phys Chem 1902;42:385.
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Flow Measurement and Instrumentation journal homepage: www.elsevier.com/locate/flowmeasinst
Calibration of electrodiffusion probes for turbulent flow measurements A. Dib a,b,n, S. Martemianov b,nn, L. Makhloufi a, B. Saidani a Laboratoire d0 Electrochimie, Corrosion et de Valorisation Energétique (L.E.C.V.E), Département de Génie des Procédés, Faculté de la Technologie, Université A. Mira Bejaia, Algérie b Institut P0 , UPR 3346 CNR S, Université de Poitiers, ENSMA,- Branche FluideBâtiment Mécanique, 3ème étage, Campus Sud, du Recteur Pineau, 86022 Poitiers, France a
art ic l e i nf o
a b s t r a c t
Article history: Received 20 May 2012 Received in revised form 4 September 2013 Accepted 21 November 2013 Available online 27 December 2013
Calibration of electrodiffusion (ED) probes with respect to the wall velocity gradient measurements has been experimentally studied in a fully developed turbulent channel flow over the Reynolds number range 14000–23000. In steady state conditions, direct calibration concerning the mean wall velocity gradient can be provided using simultaneous transient and steady state diffusion limiting current measurements. Indirect calibration of the ED probes regarding the turbulent fluctuation has been undertaken using a spectral analysis performed with probes of different size and geometry. This indirect calibration method has been supported using the measurements of thermo-dependence of molecular diffusivity. In this case, only one ED probe can be used for calibration. Dynamic calibration provides necessary information about critical longitudinal dimension of ED probes. When the longitudinal dimension of the probes exceeds the critical length, the probes becomes sensitive to the normal fluctuation as well. This fact has been confirmed using different orientation of the rectangular ED probe with respect to the flow direction. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Electrodiffusion Calibration Wall velocity gradient Near wall turbulence
1. Introduction Electrodiffusion (ED) technique is well adapted for near wall hydrodynamics and mass transfer measurements [1–3]. The possible applications concern chemical engineering, drag reduction by means of surfactants, bioengineering, micro-fluidics, multiphase flows, corrosion and deposition processes [4–11]. Moreover, this method is used as an advanced tool providing unique experimental information for verification and adjustment of theoretical models and computational codes related to near wall turbulence and mass transfer [12–15]. Different methods for ED measurements have been developed in non-stationary flow conditions [16–18]. For turbulent flows, the transfer function related the power spectral density (PSD) of wall velocity gradient and current fluctuations is usually used [2,16,19–22]. Nevertheless, this approach meets the difficulties linked to the unwanted changes of experimental conditions (temperature and depolarizer concentration) and reliability of the probes due to the alteration of their surface and geometry caused by gloating, polishing,
n Corresponding author at: Laboratoire d0 Electrochimie, Corrosion et de Valorisation Energétique (L.E.C.V.E), Département de Génie des Procédés, Faculté de la Technologie, Université A. Mira Bejaia, Algérie. Tel.: þ213 34 21 57 04; fax: þ213 34 20 51 93. nn Principal corresponding author. Tel.: þ 33 5 49 45 39 04; fax: þ33 5 49 45 35 39. E-mail addresses: dib_hafi[email protected] (A. Dib), [email protected] (S. Martemianov).
0955-5986/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.flowmeasinst.2013.11.008
bad implantation and aging. Thus, the calibration of ED probes is necessary. Calibration of ED probes with respect to time average data can be provided directly using the Lévêque method and additional control of the calibration characteristics, such as active surface area, can be achieved using a voltage step transient method [23,24]. However, the significant difference between active surface area obtained from Leveque and Cottrell methods meet the ambiguity related to the transfer function which requires knowledge of the probe dimension. It should be noted that the direct calibration of ED probes regarding to the turbulent fluctuation measurements is not possible due to inexistence of the etalon; then the problem is still open. In this paper the calibration of ED probe0 s for measuring spectra of dynamic fluctuations in turbulent flow is proposed by using probes of different sizes and geometry. For this purpose, a spectral analysis is performed to check its reliability. The proposed method is supported by taking in consideration the thermo dependence of ED measurements. This is an attractive way as only one ED probe can be used for calibration in this case.
2. Electrodiffusion method The ED method consists in measuring of limiting diffusion current provided by fast electrochemical reaction taking place at the surface of small probes flush mounted at the wall. The reduction of ferricyanide ions on a platinum cathode is the most popular
A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
Nomenclature probe active surface (m2) depolarizer concentration (mol/m3) diffusion coefficient of depolarizer (m2/s) hydraulic diameter of channel (m) diameter of circular probe (m) voltage (V) Faraday number, F ¼ 96485 (C/mol) frequency (Hz) transfer function limiting diffusion current (A) shape factor length of rectangular probe (m) width of rectangular probe (m) number of electrons involved in electrochemical reaction wall velocity gradient (s 1) temperature (K) time (s) mean flow velocity (m/s) hydraulic Reynolds number ð ¼ UDh =νÞ power spectra density of current fluctuations (A2/Hz) power spectra of wall velocity gradient fluctuations (Hz)
A c D Dh d E F f H IL k L l n S T t U Reh Wii Wss
Greek symbols
λ ρ
skin friction coefficient fluid flow density (kg/m3)
electrochemical reaction used for such flow diagnostics [1]: FeðCNÞ6 3 þ e ⇀FeðCNÞ6 4 :
I L ¼ ςS
;
ς ¼ knFcb D
2=3
ð1Þ
w
1=3
ð2Þ
where w ¼d, A ¼ π ðd=2Þ , k¼ 0.686 for a circular ED probe; w¼l, A¼ lL, k ¼0.807 for a rectangular ED probe in perpendicular position and w¼L, A ¼lL, k¼0.807 for a rectangular ED probe in parallel position. If the velocity gradient is time dependent, the Leveque solution Eq. (2) generally cannot be applied to deduce S(t) from I(t). In particular, this is the case of high frequency fluctuations when the concentration boundary layer acts as a filter and damps the current fluctuations. Thus, the current can be defined as a convolution product between the time-dependent velocity gradient and the impulse response of the system [16,22]. The impulse response of the system depends on the probe0 s geometry, mean wall velocity gradient and physico-chemical properties of the electrolyte solution. This paper is limited to the diffusion boundary layer response in the frequency domain where the power spectra1 density of the current fluctuations, Wii, is connected to that of the velocity gradient fluctuations, Wss, by 2
W ii ¼ jHðf Þj2 W ss ;
cinematic viscosity (m2/s) Lévêque coefficient (A s1/3) Cottrell coefficient (A s1/2) distance in the pressure drop (m) pressure drop (Pa) dimensionless frequency of circular probe dimensionless frequency of rectangular probe
Subscripts Lev Cot geo b h
Lévêque Cottrell geometric bulk hydraulic
Superscripts 4
dimensionless units mean value
Mathematical symbols and abbreviations R? ci R PSD ED
rectangular probe in perpendicular direction circular probe number i rectangular probe in parallel direction // power spectral density electrodiffusion probes
assuming that the axial molecular diffusion is negligible:
The mean limiting current I L is related to the mean velocity gradient S by the well-known Lévêque formula [25]: 1=3
ν ς β ΔL ΔP s s0
77
ð3Þ
where jHðf Þj is the amplitude of the transfer function of the diffusional boundary layer. This transfer function is obtained [16,22,26] by the resolution of the convective diffusion equation for the concentration fluctuations, after its linearization and by
∂C 0 ∂C 0 ∂2 C 0 ∂C þ Sx y D 2 ¼ s0x y ∂x ∂t ∂x ∂y 0 0 ∂C 1 ∂sx ∂sz 2 ∂C ; þ y s0z y ∂z 2 ∂x ∂z ∂y
ð4Þ
where s0x and s0z are the amplitude of the velocity gradient fluctuations in the axial, x, and the transversal, z, directions, and C being the mean concentration. The resolution of Eq. (4) depends on the probe0 s geometry. When the ED probe is segmented [21,27], both components of the velocity fluctuations are important and can be measured. The last term on the right-hand side of Eq. (2) deals with the normal velocity fluctuations and is usually neglected. Nevertheless, for relatively long ED probes the influence of the normal velocity fluctuations should be taken into account [22,28]. The difficulty of this approach is that the linearization assumption aforementioned is not always valid, in particular when the amplitude of the fluctuations becomes important [18,29–31]. Therefore, development of the frequency response correction using an inverse methods [18,32–37] or Sobolik0 s solution [17,38,39] leads to well resolved the issue of the dynamic behavior of the ED probe and allows correctly prediction of the wall velocity gradient with fine wall statistics analysis. In the present paper, the use of simple small ED probes (circular or rectangular) makes it possible to only preserve, in the convective diffusion equation , the longitudinal wall velocity gradient with the diffusion term in the normal direction to the flow [40,41]. In these conditions, the current spectra is related only with longitudinal velocity gradient fluctuations through Eq. (3). Moreover, most of the measured fluctuation intensities of the
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A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
diffusional current were below 12%, which means that the linearity conditions are fulfilled and that the measurements could be treated in the framework of a linear theory [29,42]. So, the linear distortion of the instantaneous velocity gradient signal, due to the mass transfer inertia of the diffusional layer, can be then examined in terms of the aforementioned transfer function [20,43]. The prediction of the later in a wide frequency range was obtained by a numerical integration of Eq. (4) [16,22,26]. The amplitude of the transfer function jHðf Þj is expressed as follows:
For rectangular ED probe: Hðs0 Þ ¼ ð1 þ 0:056s02 þ 0:0021s04 Þ 1=2 ; s0 o 6 Hð0Þ
Hðs0 Þ Hð0Þ ¼
s ¼ 2π f 0
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffiffi ð3:99 2Þ2 þ ð3:99 2 7:43 s0 Þ2 ; 2ðs0 Þ3=2 w2 DS
s0 Z 6
ð7Þ
2
Hð0Þ
s0 ¼ 2π f
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi pffiffiffiffi ð5:3 2Þ2 þ ð5:3 2 8:832 sÞ2 ; 2ðsÞ3=2 d
2
DS
2
ð6Þ
!
For circular ED probe: HðsÞ ¼ ð1 þ 0:049s2 þ 0:0006s4 Þ 1=2 ; s o 6 HðsÞ Hð0Þ ¼
ð5Þ
sZ6
ð8Þ
ð9Þ
! ð10Þ
here s0 and s are respectively dimensionless frequency of rectangular and circular probes; jHð0Þj ¼ I L =3S being the amplitude of the transfer function for the quasi-steady state conditions.
3. Experimental details The test bench is a recirculation flow loop (see Fig. 1) consisting of 460 mm length Plexiglas rectangular channel with a crosssection 15.3 mm 4 mm (aspect ratio is equal to 4). The channel is formed by a superposition of three parts. The lower part (1) contains two points for pressure drop measurements distant of 150 mm and centered on the channel axis. The higher part (2) of the channel has 20 mm diameter hole which is used for insertion of the ED probes (3). The higher and the lower parts are separated by the rectangular plate (4) giving place to a cross-section of the channel. The temperature in each experiment have been adjusted by means of a thermostat (Ecoline R312) allowing regulation of electrolyte temperature in the tank with the volume of V tank ¼ 10 l. The additional control of the temperature have been provided using a Curiolis flow-meter (ELITE, CMF050 model) introduced in hydraulic circuit outside the tank. The liquid have been pumped from the tank to a channel test section which have very small volume V channel compared to the tank volume ðV channel ¼ 0:003V tank Þ. The duration of each experiment does not exceeded 30 s, during this time the temperature variations recorded by Curiolis flow-meter have been less than 0.1 1C and the temperature in the channel has been the same as in the tank. Further details regarding the hydraulic calibration of the rectangular channel may be found in [24]. 3.1. Electrochemical probes The nomenclature of rectangular and circular ED probes fabricated from platinum sheet and wire is given in Table 1. The platinum probe is insulated by a polymeric film and glued by Epoxy resin into a large stainless steel tube using as an auxiliary electrode with the surface about 4 10 4 m2. A meticulous and progressive polishing of probes is carried out using abrasive papers with size of decreasing grains. The ED probes are flush mounted with the upper wall of the channel. The geometrical
Fig. 1. Experimental set-up.
A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
79
Table 1 Active surface area of ED probes obtained using optic, Cottrell and Lévêque method. Probes
Ageo 106 ðm2 Þ
(β 8 0.35) 106 ðA s1=2 Þ
(ς8 0.35) 106 ðA s1=3 Þ
ACot 106 ðm2 Þ
ALev 106 ðm2 Þ
c1 c2 c3 R?
0.53 0.19 0.04
26.50 9.21 2.20
9.25 3.52 0.90 3.76
0.67 0.23 0.06
0.46 0.14 0.03
0.12
0.13 0.85 ¼ 0.11
0.11 0.99 ¼ 0.11
4.60
R==
2.01
Fig. 2. Photographs of circular and rectangular ED probes.
surface of ED probes is determined by a numerical image treatment of optical microscope photographs and the rectangular ED probe can be placed either on the direction parallel or perpendicular to the mean flow (see Fig. 2). 3.2. Electrochemical system The liquid used for electrochemical flow measurements is a dilute solution containing 25 mol/m3 equimolar potassium ferro/ ferricyanide and 300 mol/m3 potassium sulphate as supporting electrolyte. The viscosity of the solution has been measured with an Ubbelholde capillary viscosimeter and the following linear equation is obtained for the temperature range T A (280–320) K with 93% accuracy [24]: T T0 νðTÞ ¼ νðT 0 Þ þ ξ1 ; ð11Þ T0
Fig. 3. Time-evolution of transient limiting diffusion current for circular and rectangular ED probes, T¼25 1C.
The experimental procedure starts with recording of Cottrell curves in order to check the state of ED probes. This method is based on the application of a voltage step (which corresponds to the diffusion plateau) and the recording of the current time evolution [46]. For very short times, the transient current can be calculated using the well known Cottrell asymptote: rffiffiffiffi D I L ðtÞ ¼ β t 1=2 ; β ¼ nFAcb : ð13Þ
where ξ1 ¼5 10 6 m2/s1 and the reference viscosity at T0 ¼298 K is equal to 10 6 m2/s. The temperature dependence of the molecular diffusion coefficient of depolarizer D in the temperature range T A (280–320) K has been obtained using the classical rotation disc electrode method [44,45]. The obtained results fit with 96% accuracy the following equation [24]: T T 0 DðTÞ ¼ DðT 0 Þ þ ξ2 ; ð12Þ T0
4. Electrodiffusion probes calibration
where ξ2 ¼4.9 10 and the reference viscosity at T0 ¼298 K is equal to 8.33 10 6 m2/s.
4.1. Calibration with respect to mean velocity gradient measurements
9
3.3. Data acquisition and signal treatment Data acquisition of the instantaneous limiting diffusion current delivered by the ED probe has been provided at the sample frequency equal to 103 Hz using a Solartron SI 1480 MultiStat potentiostat controlled by Corrwarre software. A constant polarization voltage (E¼ 0.6 V) is imposed in all experiments in order to ensure the limiting diffusion regime. Once the limiting diffusion current is recorded, the mean values of the wall velocity gradient can be determined by means of Eq. (2). Then, the current fluctuations can be treated and its power spectral density Wii can be calculated. The frequency response of the ED probes is taken into account to restore the spectra of the wall velocity gradient fluctuations Wss using the transfer function given by Eq. (3).
π
In this section we discuss the procedure of the calibration of the ED probes with respect to the mean wall velocity gradient measurements. The results of voltage step measurements are presented on Fig. 3; the good agreement of the experiment with Cottrell law is justified for all probes. Some deviation from the Cottrell behavior observed for the very short times can be explained by the double layer effects. These results justify the reliability of the sensor fabrication and can be used for the determination of the active surface area A. Our installation allows determining the mean wall velocity gradient, in the section of electrochemical measurements, by measuring of the pressure drop ΔP : S ¼ Dh
ΔP ; 4ρνΔL
ð14Þ
80
A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
therefore, it is possible to provide the direct calibration of the ED probes with respect to mean wall velocity gradient measurements using Lévêque law; the corresponding results are presented in Fig. 4. For the rectangular probe Lévêque measurements have been provided for the both positions: probe is placed parallel or perpendicular to the mean flow direction. It can be noted that all the probes satisfy correctly Lévêque law. It should be noted that the thermo-dependence of the ED measurements put on the question of the temperature effects compensation in particular for the measurements in nonisothermal conditions. Moreover, possible time variations of the active surface A and the bulk concentration cb should be taken into account. The conjoint use of Lévêque, Cottrell and rotating disc electrode measurements allows solving this problem [24]. The experimental confirmation of Lévêque and Cottrell laws give the definitive justification that the ED probes provide reliable measurements of the mean velocity gradient. Unusual transient behavior (the current IL(t) does not follow t 1=2 law) can be used as the indicator of the sensor defect [23]. Although the steady state calibration is successfully accomplished, the determination of the active surface area remains a delicate problem. The results of the
Fig. 4. Lévêque curves for circular and rectangular ED probes at T¼ 25 1C.
probe dimensions measurements obtained by means of Cottrell, Lévêque and optical methods are presented in Table 1; the significant difference for the probe surface area should be noted with respect to using method of determination. 4.2. Calibration with respect to measurements of wall velocity gradient fluctuations 4.2.1. Influence of probe geometry It is well known that the current fluctuations give an access to the turbulent fluctuations of the wall velocity gradient [1,3]. Nevertheless, in many cases, the concentration boundary layer inertia cannot be neglected and should be taken into account. Consequently, different methods have been developed and all meet the problem of sensors calibration with respect to fluctuations measurements. The common approach for the treatment of turbulent fluctuations is based on the using of the transfer function [16,22] for determination of the power spectral density (PSD), it is intimately related to knowing of the sensor geometry and the molecular diffusion coefficient of depolarizer. As we have noted above, unwanted changes in experimental conditions can occur suddenly and the active surface area may be determined with some uncertainty. So, justification of the ED method reliability regarding the turbulent fluctuation measurements is needed. For the spectra analysis of the wall velocity gradient fluctuations this justification can be achieved by using probes with different size and geometry. This justification can be achieved by simultaneous recording the wall velocity gradient fluctuations using probes with different size and geometry. Fig. 5(a) presents, as example, PSD measurements of current fluctuations obtained for Reh ¼ 14 103 using four ED probes. All the results presented in this paragraph concern the measurements provided at the fixed temperature T ¼25 1C. Obviously, the current fluctuations are very sensitive to the probes size, as well as the transfer function which links the current to the wall velocity gradient fluctuations (see Fig. 5(b)). For each probe the transfer function has been calculated using the probes dimensions obtained by means of different methods (Cottrell, Lévêque and optic). It can be noted that the dispersion of the results with respect to this uncertainty is not important (about 9%) and lies within the precision of the PSD recording. Fig. 6(a, b) presents, for two Reynolds numbers, PSD measurements of wall velocity gradient fluctuations obtained by ED probes
Fig. 5. PSD of current fluctuations (a) and transfer function (b) T ¼ 25 1C, Reh ¼ 14 103.
A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
81
Fig. 6. Dimensionless PSD of wall velocity gradient fluctuations at T ¼25 1C. (a) Reh ¼ 14 103. (b) Reh ¼ 23 103.
Fig. 7. Thermo-dependence of ED measurements obtained with circular ED probe c1 at Reh ¼14 103. (a) PSD of current fluctuations. (b) Transfer function.
with different size and geometry. The following dimensionless form of PSD is used: W ss4 ¼
Wssðf Þ : W ss ðf -0Þ
ð15Þ
The spectra of the wall velocity gradient fluctuations coincides for all the probes, this results has been confirmed for all tested Reynolds numbers. Coinciding of the PSD curves gives the justification of the reliability of the using ED probes for turbulent fluctuations measurements. This is important, because today the direct calibration of the ED probes with respect to fluctuation measurements is not possible because of the lack of a reference etalon. 4.2.2. Influence of temperature The possibility of the indirect calibration of ED probes with respect to fluctuation measurements can be supported using the thermal effects. Indeed, the transfer function depends on the
molecular diffusivity and thus on the solution temperature. From the other hand, it is reasonable to suppose that the dimensionless PSD of wall velocity gradient fluctuations via Eq. (15) is independent or at least weakly dependent on the temperature. Thus, an additional possibility for checking the reliability of ED probes regarding fluctuation measurements is possible. This possibility consists in varying of the solution temperature and recording PSD fluctuations; it is an attractive way since only one sensor can be used. As an example, Fig. 7(a) presents PSD measurements of current fluctuations recorded at different temperatures using circular ED probe C1. The temperature effects are clearly pronounced for the current fluctuation measurements. The transfer functions have been calculated taking into account the thermo dependence of the depolarizer diffusivity (Fig. 7b), and then the corresponding PSD of the wall velocity gradient fluctuations have been recovered via Eq. (3). The dimensionless PSD of the wall velocity gradient recorded at different temperatures are presented in Fig. 8(a, b). Coinciding of
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A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
the experimental curves gives the additional justification of the reliability of using ED probes for turbulent fluctuations measurements.
4.2.3. Limitations related with probe geometry It is necessary to make a distinction between the unreliability of ED probes and the interpretation of the obtained experimental data. The rectangular ED probes have proved their reliability with respect to the mean velocity gradient measurements in both positions (parallel and perpendicular) with respect to the mean flow. Fig. 9(a, b) presents, for two Reynolds numbers, the results of PSD measurements of the wall velocity gradient fluctuations obtained with the rectangular probe placed in parallel (parallel position) to the mean flow velocity position. In the same figure the PSD measurements previously obtained by circular and rectangular (perpendicular position) ED probes are presented. The significant difference of the
recorded PSD curves with respect to the probe orientation can be noted. Thus, the calibration with respect to the mean measurements is not sufficient for the fluctuation measurements. The mean current of ED probe can be sensitive to the mean flow velocity and, in the same time, the current fluctuations can be related to another hydrodynamic characteristic. The change of the probe orientation manifests itself in the increasing of the longitudinal probe size. Regarding the mean wall velocity gradient measurements the probe satisfies the Lévêque condition, but the current fluctuations do not reflect the fluctuations of the wall velocity gradient. This effect can be interpreted using the results of the theory of the ED probes which takes into account the influence of the normal velocity fluctuations [28]. According to this theory, there is a critical longitudinal length for the validity of the classical transfer function approach. This length is proportional to the viscous sublayer thickness and to the ratio between the intensity of the normal and the longitudinal velocity fluctuations
Fig. 8. Dimensionless PSD of wall velocity gradient fluctuations at different temperatures. The measurements with circular ED probe c1: (a) Reh ¼ 14 103. (b) Reh ¼ 23 103.
Fig. 9. Effect of ED probe orientation on the dimensionless PSD of wall velocity gradient fluctuations at T ¼ 25 1C. (a) Reh ¼14 103. (b) Reh ¼23 103.
A. Dib et al. / Flow Measurement and Instrumentation 35 (2014) 76–83
on the external border of the viscous sublayer. Once the longitudinal probe length is larger in comparison to this critical length, the current fluctuations is not sensitive to the wall velocity gradient fluctuations but to the normal velocity fluctuations. So, the ED probe can be sensitive to the mean velocity gradient (Lévêque law) but the current fluctuations of this probe cannot be interpreted in terms of the wall velocity gradient fluctuations. 5. Conclusion The transient voltage step method together with the steady state limiting diffusion current measurements can be used successfully for ED probes calibration with respect to mean wall velocity gradient measurements. Reliability of ED turbulent fluctuation measurements can be proved by using probes with different size and geometry. Coinciding of all wall velocity gradient PSD, within the experimental accuracy range, justifies the probe0 s reliability. The temperature effects open additional possibility for checking of the ED probes reliability. Indeed, dimensionless PSD of wall velocity gradient fluctuations is quasi independent on the temperature. Once the reliability of the ED probes is proved, a fine interpretation of PSD measurements becomes possible. In particular, verification of the Lévêque law is not sufficient for the conclusion that the current fluctuations reflect the wall velocity gradient fluctuations. This fact has been clearly confirmed using different orientation of the rectangular ED probe with respect to the flow direction. Changing the probe0 s orientation make it sensitive to normal velocity fluctuations and, thus, open additional possibilities in ED flow diagnostic. Acknowledgments This work pertains to the French Government program “Investissements d0 Avenir” (LABEX INTERACTIFS, reference ANR-11LABX-0017-01). References [1] Mitchell J, Hanratty T. A study of turbulence at a wall using an electrochemical wall shear-stress meter. J Fluid Mech 1966;26:199–221. [2] Fortuna G, Hanratty T. Frequency response of the boundary layer on wall transfer probes. Int J Heat Mass Transf 1971;14:1499–507. [3] Hanratty TJ, Campbell JA. Measurement of wall shear stress, fluid mechanics measurements. Washington: R.J. Goldstein, Hemisphere; 1983. [4] Legrand J, Legentilhomme P, Aouabed H, Ould-Rouis M, Nouar C, Salem A. Electrodiffusional determination of momentum transfer in annular flows: axial developing and swirling decaying flows. J Appl Electrochem 1991;21:1063–7. [5] Adolphe X, Danaila L, Martemianov S. On the small-scale statistics of turbulent mixing in electrochemical systems. J Electroanal Chem 2007;600:119–30. [6] Alekseenko SV, Markovich DM, Evseev AR, Bobylev AV, Tarasov BV, Karsten VM. Experimental study of liquid distribution in a column with a structured packing. Theor Found Chem Eng 2007;41(4):417–23. [7] Boutoudj M, Ouibrahim A, Barbeu F, Deslouis C, Martemianov S. Local shear stress measurements with microelectrodes in turbulent flow of drag reducing surfactant solutions. Chem Eng Process: Process Intensif 2008;47:793–8. [8] Blel W, Le Gentil-Lelièvre C, Bénézech T, Legrand J, Legentilhomme P. Application of turbulent pulsating flows to the bacterial removal during a cleaning in place procedure. Part 1: experimental analysis of wall shear stress in a cylindrical pipe. J Food Eng 2009;90:422–32. [9] Huchet F, Comiti J, Legentilhomme P, Solliec C, Legrand J, Montillet A. Multiscale analysis of hydrodynamics inside a network of crossing minichannels using electrodiffusion method and PIV measurements. Int J Heat Fluid Flow 2008;29:1411–21. [10] Kashinskii O, Kaipova E, Kurdyumov A. Application of the electrochemical method for measuring the fluid velocity in a two-phase bubble flow. J Eng Phys Thermophys 2003;76(6):1215–20. [11] Barbier F, Alemany A, Martemianov S. On the influence of a high magnetic field on the corrosion and deposition processes in the liquid Pb-17Li alloy. Fusion Eng Des 1998;43:199–208. [12] Parys H, Tourwé E, Breugelmans T, Depauw M, Deconinck J, Hubin A. Modeling of mass and charge transfer in an inverted rotating disk electrode (IRDE) reactor. J Electroanal Chem 2008;622:44–50.
83
[13] Legrand J, Martemyanov SA. Turbulent mass transfer in small radius ratio annuli. J Appl Electrochem 1994;24(8):737–44. [14] Martemianov S, Okulov V. Mass transfer ambiguities in swirling pipe flows. J Appl Electrochem 2002;32(1):25–34. [15] Pallares J, Grau F. Frequency response of an electrochemical probe to the wall shear stress fluctuations of a turbulent channel flow. Int J Heat Mass Transf 2008;51:4753–8. [16] Deslouis C, Gil O, Tribollet B. Frequency response of electrochemical sensors to hydrodynamics fluctuations. J Fluid Mech 1990;215:85–100. [17] Sobolik V, Wein O, Cermak J. Simultaneous measurement of film thickness and wall shear stress in wavy flow of non-Newtonian liquids. Collect Czechoslov Chem Commun 1987;52:913–28. [18] Rehimi F, Aloui F, BenNasrallah S, Doubliez L, Legrand J. Inverse method for electrodiffusional diagnostics of flows. Int J Heat Mass Transf 2006;49:1242–54. [19] Ambari A, Deslouis C, Tribollet B. Frequency response of the mass transfer rate in a modulated flow at electrochemical probes. Int J Heat Mass Transf 1986;29:35–45. [20] Nakoryakov VE, Kashinsky N, Kozmenko BK. Electrochemical method for measuring turbulent characteristics of gas-liquid flows, measuring techniques in gas–liquid two phase flows. In: IUTAM symposium, Nancy, France; 1983. p. 695–721. [21] Deslouis C, Huet F, Gil O, Tribollet B. Spectral analysis of wall turbulence with a bicircular electrochemical probe. Exp Fluids 1993;16:97–104. [22] Skurygin E, Martemianov S, Vorontyntsev M, Grafov B. Computing the frequency characteristics of turbulent diffusion layers at microelectrodes. Sov Electrochem 1989;25:685–8. [23] Sobolik V, Tihon J, Wein O, Witcherle K. Calibration of electrodiffusion friction probes using a voltage-step transient. J Appl Electrochem 1998;28:329–35. [24] Dib A, Martemianov S, Makhloufi L, Saidani B. Thermo-compensation of electrodiffusional wall velocity gradient measurements. Int J Heat Fluid Flow 2008;29:219–28. [25] Lévêque M. Transmission de chaleur par convection. Ann des Mines 1928;13:381–412. [26] Nakoryakov VE, Burdukov AP, Kashinsky ON, Geshev PI. Electrodiffusion method of investigation into the local structure of turbulent flows. Novosibirsk: V.G. Gasenko; 1986. [27] Sobolik V, Wein O. Three-segment electrodiffusion probes for measuring velocity fields close to a wall. Exp Fluids 1990;9:43–8. [28] Martemyanov SA, Petrovskiy NV, Grafov BM. Turbulent pulsations of the microelectrode limiting diffusion current. J Appl Electrochem 1991;21 (12):1099–102. [29] Kaiping P. Unsteady forced convective heat transfer from a hot film in nonreversing and reversing shear flow. Int J Heat Mass Transf 1983;26:545–57. [30] Mao Z, Hanratty TJ. Analysis of wall shear stress probes in large amplitude unsteady flows. Int J Heat Mass Transf 1991;34(1):281–90. [31] Mao ZX, Hanratty TJ. The use of scalar transport probes to measure wall shear stress in a flow with imposed oscillations. Exp Fluids 1985;3:129–35. [32] Mao Z, Hanratty TJ. Application of an inverse mass transfer method to the measurement of turbulent fluctuations in the velocity gradient at the wall. Exp Fluids 1991;11:65–73. [33] Maquinghen T. Métrologie tridimensionnelle instationnaire à l0 aide de la méthode polarographique [Ph.D. thesis]. Thèse de doctorat, Université de Valenciennes; 1999. [34] Berrich E, Aloui F, Legrand J. Experimental validation and critical analysis of inverse method in mass transfer using electrochemical sensor. Exp Therm Fluid Sci 2013;44(0):253–63. [35] Berrich E, Aloui F, Legrand J. Inverse method for the dynamical analysis of wall shear rates using three-segment probes in parallel plate rheometer. Chem Eng Sci 2011;66(17):3969–78. [36] Zidouh H, Labraga L, William-Louis M. Unsteady wall shear stress in transient flow using electrochemical method. J Fluids Eng 2009;131(5):1 http://dx.doi. org/10.1115/1.3112387. [37] Keirsbulck L, Labraga L, Gad-el Hak. Statistical properties of wall shear stress fluctuations in turbulent channel flows. Int J Heat Fluid Flow 2012;37(0):1–8. [38] Huchet F, Legentilhomme P, Legrand J, Montillet A, Comiti J. Unsteady flows in milli- and microsystems: analysis of wall shear rate fluctuations. Exp Fluids 2011;51(3):597–610. [39] El Hassan M, Assoum H, Sobolik V, Vétel J, Abed-Meraim K, Garon A, et al. Experimental investigation of the wall shear stress and the vortex dynamics in a circular impinging jet. Exp Fluids 2012;52:1475–89. [40] Ling S. Heat transfer from a small spanwise on an insulated boundary. J Heat Transfer Trans ASME 1963:230–6. [41] Mollet L, Dumargue PMD, Bodiot D. Calcul du flux limite de diffusion sur une microelectrode de section circulaire. Equivalence avec une électrode de section rectangulaire. vérification expérimentale dans le cas du disque toumant en régime laminare. Efectrochim Acta 1974;19:841–4. [42] Rode S, Midoux N, Latifi M, Storck A, Saatdjian E. Hydrodynamics of liquid flow in packed beds: an experimental study using electrochemical shear rate sensors. Chem Eng Sci 1994;49(6). [43] Deslouis C, Gil O, Tribollet B. Frequency response of electrochemical sensors in a cone-and-plate modulated flow. Int J Heat Mass Transf 1990;33(11):2525–32. [44] Coeuret F, Storck A. Eléments de génie électrochimique. Paris: Tec et Doc Lavoisiers; 1984. [45] Levich V. Physicochemical hydrodynamics. Englewood, USA: Prentice-hall; 1962. [46] Cottrell F. Courant de réponse à une polarization galvanique traité comme un problème de diffusion. Z Phys Chem 1902;42:385.