Experimental Thermal and Fluid Science 44 (2013) 512–519

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Experimental Thermal and Fluid Science

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Simultaneous film thickness measurement and wall temperature assessmentby Low-Coherence Interferometry

Nicolas Borgetto, Frédéric André, Cédric Galizzi ⇑, Dany EscudiéUniversité de Lyon, CNRS, INSA-Lyon, CETHIL, UMR5008, F-69621 Villeurbanne, FranceUniversité Lyon 1, F-69622 Villeurbanne, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 2 February 2012Received in revised form 17 July 2012Accepted 3 August 2012Available online 24 August 2012

Keywords:Low-Coherence InterferometryFalling liquid filmSurface waveThickness measurementWall temperature assessment

0894-1777/$ - see front matter � 2012 Elsevier Inc. Ahttp://dx.doi.org/10.1016/j.expthermflusci.2012.08.01

⇑ Corresponding author. Address: Bâtiment Sadi C69621 Villeurbanne Cedex, France. Tel.: +33 04 72 4311.

E-mail address: cedric.galizzi@insa-lyon.fr (C. Gali

Film thickness is one of the most important parameters characterizing a liquid film flowing along a wall.The wall’s surface temperature intervenes in heat transfer within the film and, by means of it, modifiesthe film’s flow properties. The measurement of film thicknesses and wall temperatures allows us to studythe coupled physical mechanisms in anisothermal configurations, and can be achieved using the nonintrusive technique of Low-Coherence Interferometry (LCI). This method was initially selected since itcan be applied in our main specific configuration, which involves a wall coated with a liquid fuel filmas it interacts with a reactive flow. The majority of existing techniques are ill-adapted to such a config-uration. Prior to analysing such a complex case, LCI was first applied in a simple, non reactive case. Thissituation is studied in the present work and consists of a vertical wall over which a liquid film is flowingfreely. The principle of LCI is briefly explained, as well as how waves on the free surface of the film canimpact its measurement. The experimental setup is described and used to study several liquid film flowdynamics. Results showed that the measured mean thickness of the film was in accordance with the Nus-selt model. In addition, two distinct regimes were identified, conformally to numerical analysis from lit-erature. Regarding the wall temperature assessment by LCI, the comparison of the results with thoseobtained using another technique demonstrated the feasibility of this type of non intrusive estimation,as it was possible to achieve a 2 �C deviation between the two approaches. This is promising becausemeasuring this parameter remains difficult. Finally, the analysis of the results made it possible to synthe-size the advantages and limitations of the LCI technique when applied to two-phase flow systems.

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1. Introduction

In many industrial processes, liquid films deposited and flowingon walls are encountered. This situation is sometimes a key elementof the process, as in coating technologies, or a consequence of thetechnique used to inject the liquid into the process, as in some com-bustion applications. Indeed, in gas turbines using Lean PremixedPrevaporized (LPP) systems and Gasoline Direct-Injection (GDI)piston engines [1], which are two common techniques to reducefuel consumption and CO2 emission, liquid fuels are sprayed insidethe combustion chamber and can be deposited on the wall. Thisleads to the coexistence, inside the chamber, of a combustible liquidfilm on the walls and a flame in its vicinity. This configuration is theseat of coupled physical and chemical interactions between the filmand the flame which have effects on combustion efficiency as wellas on pollutant emission. For these reasons, specific studies are

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required in order to deepen our knowledge about how liquid filmsand flames interact within confined systems.

A small number of numerical approaches [2] have been carriedout for this purpose, but they are generally lacking in experimentalvalidations. Thus, an experimental device has been developed withthe following objectives: (1) to provide a better understanding ofthe mechanisms encountered during wall/film/flame interaction(for which very few data are available in actual situations suchas engines, due to the confinement of the combustion medium)and (2) to provide reference databases to be used as benchmarksfor the assessment of numerical models. The experimental setupconsists of a vertical plane wall over which a liquid film can be gen-erated. The film can flow down and interact with an upward reac-tive stream consisting of a rod stabilized oblique V-shaped flame(in a lean premixed methane/air mixture). The film is obtainedby injecting the liquid through a porous medium inserted withinthe wall. Due to the overall complexity of the problem, it wasdecided to adopt a progressive approach (with an increasing de-gree of intricacy). One of the first steps of the work consists instudying and assessing the experimental setup in a simple, nonreactive configuration. This is the aim of the present work.

Nomenclature

AbbreviationsBS Beam SplitterC CollimatorLCI Low-Coherence InterferometryLCLS Low-Coherent Light SourceMM Mobile MirrorX, Y acquired data

Latin symbolsd geometrical thickness (m)D beam diameter (m)g gravitational acceleration (m s�2)h0 film thickness from Nusselt solution (m)I intensityL length scale (m)M interfacenG group refractive indexOPL Optical Path Length (m)q0 volumetric flow rate per unit width (m2 s�1)r reflection coefficientR reflectivityReT Reynolds numbert time (s)t transmission coefficientT temperature (�C)uN velocity at the film/gas interface (m s�1)

We Weber number

Greek symbolsc surface tension coefficient (N m�1)d0 mobile mirror position (m)d⁄ Takeshi [10] parameterk wavelength (nm)k0 average wavelength of light source (nm)l dynamic viscosity (Pa s)m kinematic viscosity (m s�2)q density (kg m�3)r standard deviationv rate of successful measurements

Subscriptsa airf filmi index iin inletref referencet timeTC thermocouplew wallk wavelength

Fig. 1. The principle of the Michelson interferometer with a LCLS and a bi-layersample composed of transparent wall and film–angles are represented in order tofollow the light beams.

N. Borgetto et al. / Experimental Thermal and Fluid Science 44 (2013) 512–519 513

Among the characteristics of liquid films flowing on verticalwalls, two parameters are of particular interest. The first of theseis the thickness of the film [3]. This quantity can be determinedusing various techniques [4], among which the most suitable isLow-Coherence Interferometry [5], as retained in this work. Indeed,due to the proximity of the flame, measurements must be takenfrom the wall side, thus limiting the number of characterizationtechniques available [6]. The second parameter of particular inter-est to us is the temperature of the interface between the film andthe vertical wall. This quantity is linked to heat fluxes and it is notoften available since its measurement by probes would modifythe flow in the vicinity of the wall, which would have an impacton measured temperatures. These two parameters are coupled withheat and mass transfer phenomena: for instance, thermal energytransferred by the wall to the liquid film plays a key role in its evap-oration and therefore in its thickness. Their simultaneous determi-nation can thus provide an insight into the fundamental physicalprocesses involved during wall/film/gas flow interaction.

The aim of the present paper is to describe how the LCI tech-nique can be used in order to determine film thicknesses and thetemperatures of vertical solid surfaces in contact with liquid films.In the first part (Section 2), the basic principles of the LCI tech-nique, film thickness and wall temperature measurements are gi-ven. Next, several specific aspects of liquid films flowing on avertical plate are provided in Section 3. A simple model is proposedto investigate the reliability of the technique. This study also pro-vides a better understanding of the possible sources of error. InSection 4, the experimental setup is detailed and the procedureused to measure and analyse data is presented and applied tothe study of heptane liquid films deposited on a vertical silica wall.Results in terms of film thickness and wall temperature measure-ments are provided and discussed in Section 5, along with theircorresponding limitations and accuracies.

2. The principles of film thickness measurement and walltemperature assessment by LCI

The ‘‘object’’ studied in this work is composed of two main com-ponents: a plane vertical wall and a thin liquid film that flowsfreely along it. The film is in contact with the wall on one sideand with the air on the other. The objective of the following sectionis to describe briefly what Low-Coherence Interferometry is andhow it can be used to characterize the thickness of the liquid film.

The LCI principle shares many similarities with the MichelsonInterferometer, as shown schematically in Fig. 1. This techniqueuses several key optical elements: a Low-Coherent Light Source(LCLS), a Beam Splitter (BS) and a detector, which are used toestimate the optical path difference between two branches of an

Table 1Optical path lengths for the interfering optical signals.Notations are those of Fig. 1.

Optical PathLength (OPL)

Value

L01 L0 + 2L + L0

L02 L0 + 2L + L0 + 2nGwdw

L03 L0 + 2L + L0 + 2nGwdw + 2nGfdf

L04 L0 + 2L + L0 + 2d0

Fig. 2. Interferogram patterns obtained with Gaussian LCLS and illustration of theinterface location and the interference signal envelope.

514 N. Borgetto et al. / Experimental Thermal and Fluid Science 44 (2013) 512–519

optical setup: one is used as a reference and the other is the samplewe want to characterize. The BS separates the intensity of the lightsource into two (theoretically equal) components: one of thesetravels along the reference arm, is reflected by the Mobile Mirror(MM) and goes back to the BS; the other propagates within thesample, reflected by the free surface of the film (which acts as afixed mirror) and then sent back into the medium and the BS.For the purpose of our experiments, the sample is composed of atransparent bi-layer medium. One layer corresponds to the wall(indicated by subscript w), the other is the liquid film (subscriptf). Each of them is characterized by a group refractive index nG

and a geometrical thickness d. They are separated from each otherby interfaces, which play the role of reflective surfaces (Mi)i=1,2,3.When travelling along the sample branch, the light is reflectedby each ‘‘mirror’’ encountered in its path. This produces three opti-cal signals with Optical Path Lengths (OPLs) which are differentfrom that of the initial source and separated from it by OPL L01,L02 and L03 respectively (see Table 1). When they reach the detec-tor, these signals interfere with the reference signal which hastravelled along a total optical path L04, within the interferometer,that is a function of the displacement of the moving mirror. Thisproduces an interference pattern on the detector, which can beused to determine the differences in OPLs and, consequently, thedistance between all the interfaces in the sample (assuming thatthe group refractive indexes of the media are known). This opticalpattern is called an interferogram.

Fig. 2 illustrates a typical interferogram which can be obtainedusing this technique. It displays two interference patterns whichcorrespond to the two interfaces that encompass the liquid filmlayer (M2 and M3). The envelope of interference signals is closelyrelated to the characteristics of the Low-Coherence Light Source(a Gaussian emission spectrum was chosen for this example).The locations of peaks of interference (as a function of the mobilemirror displacement d0) and their magnitudes depend on the thick-ness and the optical properties of the wall and film layers. Maximaof the interference patterns correspond to each Mi interface and theOPL between them can be determined (from the knowledge of d0).OPLf, which represents the OPL associated with the liquid film, canbe obtained by subtracting the positions of the two interferencemaxima before and after it.

The exploitation of reflected light to measure temperaturedependencies of the refractive indexes of liquids was proposedby Kim and Su [7]. Their method is based upon the fact that thereflective properties of the interface are connected to the refrac-tive indexes, which are functions of the temperatures of the mediaon each side of the interface. We could apply a similar techniquein our study by measuring the magnitude I2 corresponding tothe maximum of the interferogram, in two situations: one isused as a reference and is obtained without liquid on the wall;the second corresponds to a value with the film (further detailsare provided hereafter). In our application, OPLf and I2 measure-ments thus enable us to estimate respectively the liquid filmthickness and wall temperature. The steps to achieve this aresummarized below:

Step 1: The film thickness is a function of its group refractiveindex nGf and the measured OPLf that can be simply estimatedas follows:

df ¼OPLf

nGfð1Þ

Step 2: Wall temperature can simultaneously be determined bymeasuring I2 which is closely related to the amplitude of theelectromagnetic fields in the two arms of the interferometric

device. In the case of a planar wave under normal incidence,the transmission and reflection coefficients are given, respec-tively for air/wall and wall/film interfaces, as [8]:

taw ¼2na

na þ nw¼ na

nwtwa ð2Þ

rwf ¼nw � nf

nw þ nfð3Þ

where na, nw and nf are the refractive indexes of the air, the walland the liquid, defined at the temperature of the interface betweenthe fluid (air or liquid) and the wall. The amplitude of I2 is propor-tional to the square root of the reflectivity of the wall/film interface(Rwf ¼ r2

wf ) [5], and depends on its temperature. Other unknownparameters are also involved in its calculation. Nevertheless, if a ref-erence intensity I2 ref is recorded under isothermal conditions with-out a film, it is then proportional to the reflection coefficient of thewall/air interface |rwa|. The following ratio:

I2

I2 ref¼ jrwf jjrwaj

ð4Þ

provides a way to attain our objective if unknown parameters (lightsource power, radiative properties of the wall, optical response ofthe experimental setup) are assumed to remain constant betweenthe two experiments.

These equations are useful as they describe how the LCI tech-nique can be used to measure liquid film thickness and tempera-ture. Nevertheless, several additional theoretical elements relatedto the liquid film flowing on a vertical plate configuration are re-quired in order to estimate the validity of the model proposed. In-deed, when a liquid film material flows on a wall: (1) its thicknessdepends on its thermophysical properties and (2) the liquid/gasinterface is not perfectly flat, as assumed in the model presentedin this section. These sources of potential errors must be analysed

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in order to understand their influences on the LCI measurementtechnique in the context of this study.

3. A liquid film flowing on a vertical plate and consequences onits thickness measurement by LCI

The stability theory demonstrates that a liquid film flowing on avertical flat plate is unstable for any Reynolds number [9]. It alsopoints out two distinct regimes that can be classified with respectto the following parameter d�, proposed by Takeshi in Ref. [10]. Itcompares inertial and gravitational effects with the kinematic vis-cosity and the surface tension of the fluid:

d� ¼ ReT

We1=3 ð5Þ

ReT and We are respectively Reynolds and Weber numbers de-fined as:

ReT ¼uNh0

mð6Þ

We ¼ cqgh2

0

ð7Þ

In Eqs. (6) and (7), q is the liquid density in kg m�3, c its surfacetension coefficient (N m�1), (m = l/q) is the kinematic viscosity(m2 s�1) of the fluid and g the gravitational acceleration(g = 9.81 m s�2). h0 and uN are respectively the theoretical thick-ness of the film (m) and the velocity (m s�1) at the film/gas inter-face. They can be derived from the Nusselt formulas for smoothlaminar film flow [11]. This model makes it possible to link the vol-umetric flow rate per unit width q0 with the flat film thickness h0:

q0 ¼gh3

0

3mð8Þ

This theoretical approach can be used to assess the validity ofthe experimental approach, from the optical setup up to the anal-ysis technique, in this simple case.

Takeshi [10] showed that: (i) when d� � 1, the flow has a behav-ior that can be represented as the Nusselt flat film solution; (ii)when d� is close to unity, or takes values higher than 1, magnitudesof waves at the film surface increase. In this case, the assumptionof a parallel flat liquid/gas interface (called M3 in Fig. 1) can beerroneous and lead to experimental problems resulting in a de-crease in thickness measurement accuracy.

Indeed, when the liquid/gas interface (M3) is not rigorously par-allel to the wall, due to instabilities, the direction followed by the

Fig. 3. Potential consequences of a wavy surface on liquid film thicknessmeasurement by LCI.

light reflected by this surface strongly depends on its shape. Fur-thermore, the film flows along the wall and the shape of the li-quid/gas interface evolves with time. The potential consequencesof this phenomenon on liquid film thickness measurement by LCIare illustrated in Fig. 3 for four situations:

(a) This case (flat liquid film) was introduced together with theprincipleof the LCI technique in Section 2. As(Mi)i=1,2,3 interfacesare parallel, the beam diameter D is always seen by the detector.Each MM scan provides a film thickness measurement.

(b) When the liquid/gas interface is not parallel to the two othersurfaces, the reflected signal is only partially detected,depending on the angle between the incident beam andthe normal to the liquid/gas interface. This decreases theintensity of the interference patterns as well as the corre-sponding signal-to-noise ratio. In some cases, no reflectedlight reaches the detector. In this last situation, film thick-ness measurement is impossible. If thickness fluctuationsbecome important and the optical setup is highly selectiveto reflected angles, only a small amount of the total numberof scans will be collected by the optics and seen by thedetector. We have thus introduced the following rate of suc-cessful measurements v:

F

v ¼ Number of MM scans providing thicknessTotal scans number

ð9Þ

to provide a qualitative factor for the deformation of the free sur-face of the film.

(c) When the length scale kf of this surface (schematized by asinusoidal function) becomes smaller than the light beamdiameter D, measurements cannot be considered as local.Indeed, only zones where the M3 interface is quasi-parallelto wall ones are detected. They induce an optical path lengthheterogeneity and undesired interferences along the lengthscale Lk that can make representative measurement difficult.

(d) The last case illustrates two interfaces M3 at different timest1 and t2. If the scan speed is too low, in comparison with thefilm wave velocities, the interferogram will take intoaccount the variation of the shape of the free surface. Thiscan produce interferences along the scale Lt that are linkedprincipally to the amplitudes of the waves.

A dedicated experimental device was developed to enable localand instantaneous liquid film thickness measurement. This deviceis described in the next section.

ig. 4. Optical LCI apparatus and vertical wall coated with liquid film.

516 N. Borgetto et al. / Experimental Thermal and Fluid Science 44 (2013) 512–519

4. Experimental device and procedures

Fig. 4 represents the optical device developed in this work. LCLS isa Super luminescent LED (SLED, DL-CS3159A, Denselight). It has aGaussian spectrum centred at k0 = 1310 nm with about 80 nm fullwidth at half maximum. This source can output 15 mW opticalpower with a coherence length lower than 20 lm. The beam splitterconsists of a 2 � 2 optical fibre coupler. After passing through thetwo branches of the setup, signals are collimated by two opticallenses C1 and C2 (F240APC-C, Thorlabs). This makes the device selec-tive to incidence angles and produces a circular output beam (with a1.4 mm diameter). The reference arm consists of an infrared mirrorattached to a translation device (M-682.174 PILine, PI) driven by pie-zoelectric motorization, which allows high accelerations (up to 10times the gravitational one) and speeds (as high as 500 mm s�1).The detector is a 200-kHz Photoreceiver (2011-FC, New Focus).The location of the translation stage, Xi, as well as the interferenceintensity, Yi, are recorded simultaneously by an acquisition system(NI PXI 6259, National Instruments) connected to a PC. The com-puter is also used to post-treat the experimental data.

The bi-layer sample is composed of the vertical wall and the li-quid film. The wall is made of fused silica (HPFS� Fused Silica Stan-dard Grade, Corning) and was manufactured by Schott. Its mainspecifications were a thickness of 13 mm, a surface planarityexceeding 5 lm and parallelism tolerance of 20 lm. Surface rough-ness was imposed at less than 0.6 lm to avoid scattered reflection.Fused silica is nearly transparent at source maximum intensity (atransmission function higher than 95% at 1310 nm). The liquid(chosen as a combustible liquid for the general purpose of the pro-ject in which the present study is included) is heptane with 95%purity, provided by Carlo Erba Reactifs-SDS. It is delivered contin-uously by a micro gear pump (Micropump) with a displacementvolume of 0.092 mL per revolution. It is driven by an electric motor(Ismatec) with a rotating speed in the [50; 5000] revolutions perminute range. This mechanical device is disconnected from theoptical elements and wall so as to avoid the transmission of vibra-tions. The liquid is injected onto the vertical wall through a porousmedium consisting of an 80 mm � 80 mm square of silica.

All of the optical elements were selected to be as close as pos-sible to the theoretical requirements. Indeed, in terms of:

(a) The stability of the radiation emitted by the LCLS: drifts oflight source power are considered negligible as it is suppliedby a stable current source and is connected with a tempera-ture controller (±0.1 �C).

(b) The thermophysical properties of the media: air and wall(fused silica material) thermo-optic coefficients are small atk0 and wall thermal expansion is negligible (�5.7 � 10�7 �C�1

from 0 to 200 �C) which limits possible OPL variations withtemperature.

(c) The stability of the detection device: detector response canvary during an experiment because of temperature and elec-trical fluctuations. Nevertheless, preliminary tests haveshown that variations in terms of measured intensities cor-respond to a maximum absolute error of around ±1.5 �C forwall temperature measurement under stationary conditions.

This setup was used to measure liquid film thicknesses and walltemperatures. This required the definition of a specific procedurewhich is detailed in the following subsections.

4.1. General procedure

A full measurement campaign involves several steps, from thesetting of the optical device and data acquisition to numericalpost-treatments. These steps are detailed hereafter:

(1) The collimators C1 and C2 were aligned so that the lightreflected by MM and the interfaces of the plate was sentback to the interferometer. The distance between C1 andthe wall is 10 cm. The studied zone is located 3 cm belowthe porous medium outlet and at the centre of the plate.

(2) The MM translation stage was driven following a sinusoidalprofile with a frequency of 1 Hz and an amplitude of 2 mm.The sampling frequency was more than five times that of thefringe pattern (which is a function of the translation stagespeed and k0). This corresponds to the acquisition of 105 -samples per second at the maximum speed of the MM(around 10 mm s�1). An analogue band-pass filter was usedto filter the DC component of the interferogram at the detec-tor. Band limits were set, after tests, between 3 and 100 kHz.During each scan, the synchronized acquisition of Xi (MMposition) and Yi (detector intensity) was performed.

(3) After their acquisition, the interferograms were automati-cally post-treated by a computer program developed in theLabview programming environment. The analysis uses a fit-ting technique to adjust Gaussian functions to the envelopeof the peaks in the interference signal. Each adjustment pro-vides three parameters for the peaks.

– Their centres, which are later used to determine the thick-

ness of the liquid film as indicated in the previous section.– Their amplitudes, I2, which can enable us to estimate the

temperature of the interface between the film and thewall.

– Their standard deviations, which provide a quantitativeparameter to check if the interference peak shape relatedto the M3 interface is modified by the presence of surfacewaves on the liquid film.

Repeatability tests were carried out on wall interface locationsand interference magnitudes in order to study the possible influ-ence of undesirable effects such as vibrations on the measure-ments. With 100 interferograms, the results showed smallstandard deviations: in terms of interface locations, they werearound 1 lm; relative errors on detector maximum intensitieswere lower than 1%.

Additional specific steps were required to measure liquid filmthickness and assess wall temperature.

4.2. Procedure related to thickness measurement and uncertainty

Each measurement consists of between a hundred and a thou-sand scans from which we can determine, using a statistical meth-od, an average film thickness, its standard deviation and thevalidation rate of measures v.

Errors on film thickness are due to several contributing factorsincluding: the effects of optical parameters and of temperaturevariation as well as errors related to the apparatus and the numer-ical post-treatment technique. The influence of thermal gradientscan be considered as negligible in the case of heptane liquid asnoted in Ref. [6]. In order to assess other factors, a simulation toolwas developed to generate numerical interferograms. These weresubsequently analysed with the same program used to post-treatthe experimental interferograms. It was then possible to measurethe absolute deviation of OPLf induced by certain parameters suchas light source properties, the refractive index and wall absorptionvarying with wavelength and signal-to-noise ratio associated withthe interference peak intensity. The overall contribution obtainedin the most unfavorable conditions was 4 lm, which was consid-ered to be the absolute uncertainty of OPLf measurement.

The use of Eq. (1) to determine the film thickness requires anestimate of the group refractive index of the liquid, nGf. Thisquantity depends on the temperature [6] of the film, which is

N. Borgetto et al. / Experimental Thermal and Fluid Science 44 (2013) 512–519 517

determined experimentally by a thermocouple located at the flowinlet of the porous medium, assuming an isothermal flowing film.Eqs. (5)–(8) require values for the density, kinematic viscosity andsurface tension coefficient of the liquid which also depend on tem-perature. These were calculated with data from Ref. [12].

Estimates of liquid flow rates are needed to enable comparisonsbetween the experimental data for film thickness and the theoret-ical formula given by Eq. (8). We have calibrated the system used toinject the fluid into the porous medium by a weighing technique.The density of the liquid was estimated from its temperature andused to calculate the volumetric flow rate associated with the massof liquid recovered at the base of the wall. Repeatability tests haveshown that the maximum absolute deviation was around1.0 mL mm�1. Dividing the volumetric flow rate by the width ofthe film provides q0. A variation of this parameter modifies the filmthickness h0 calculated with Eq. (8). By means of this procedure, theLCI technique could be assessed by comparing experimental andtheoretical data (from the Nusselt solution, as presented previ-ously) over a wide range of film thicknesses and liquid flow dy-namic behaviors, characterized here by d� (cf. Eq. (5)).

4.3. Procedure for checking the pertinence of wall/liquid interfacetemperature assessment

As mentioned earlier, an accurate measurement of the tempera-ture of the walls in contact with the liquid film is difficult whatevertechnique is used. This makes the assessment of measurements ob-tained by LCI against those obtained using other approaches tricky.However, it is possible to investigate the reliability of such measure-ments by comparing them with data obtained using a thermocouple.Obviously, the presence of the thermocouple (in contact with thewall) may perturb the liquid flow as well as the temperature weare trying to measure. Nevertheless, this approach should providesome indications as to whether or not LCI is able to achieve theobjective of the study.

A heat exchanger was used to preheat the liquid and control itstemperature before its injection into the porous zone of the wall.Temperatures in the range of [10; 50] �C were selected. The higherthe difference between the mean temperature of the fluid and thewall, the higher the difference between the values given by LCI andthe thermocouple should be. Indeed, when the thermal gradientincreases within the film, its impact on the thermocouple measure-ment rises. We have checked this effect under stationary condi-tions. Three liquid film thicknesses and flow regimes were studied.

Thus, experiments with and without a liquid film flowing on thevertical plate were conducted to determine I2 and I2 ref respectively.As indicated in Eqs. (3) and (4), temperature assessment from thesequantities requires an estimate of the refractive index of the variousmedia at the wavelength used by LCI and as a function of their tem-perature. The absolute value of the wall’s thermo-optic coefficient(@nw/@T)P is small (�10�5 �C�1) in comparison with that of the hep-tane (�5 � 10�4 �C�1) at k0. Thus, nw was calculated using Corningdata and considered to be fixed at 1.44680. The refractive index ofair na was also considered to remain constant at 1.00027 [13] overthe temperature and wavelength ranges involved in these experi-ments. The refractive index of heptane was evaluated at k0 accord-ing to the temperature following data from Refs. [6,14]. Usingthese values in Eqs. (3) and (4) enabled us to find parameter Tw.

The results, based on the theoretical and experimental elementsdescribed previously are reported in the next section.

Fig. 5. Comparison of experimental data with the Nusselt solution (Eq. (8)). (a)Measured time-averaged thickness df as a function of calculated thickness h0. (b)Relative deviation between experiment and theory according to h0.

5. Results and discussions

Experimental interferograms were acquired following theprocedure detailed in Section 4. Their analysis has shown that

standard deviations given by Gaussian adjustments performed onthe peaks associated to the M3 interface remain generally closeto the others (M1 and M2). In line with previous comments (cf. Sec-tion 3), thicknesses measured by the LCI apparatus we developedcould be assumed to be local and instantaneous (viz. not perturbedby surface waves at the liquid/gas interface).

Theoretical and experimental methods were assessed againsteach other. Fig. 5a shows the mean value of measured (df) and the-oretical (h0, calculated using Eq. (8)) thicknesses as functions of thevolumetric flow rate per unit width q0. Data remain close to theline df ¼ h0. An absolute repeatability error of 1 mL min�1 wasdetermined experimentally for the liquid flow rate. The corre-sponding absolute errors are given in Fig. 5a with their associatederror bars. These errors, and particularly those related to liquidflow rates, indicate that the thinner the film, the higher the relativeerror between the theoretical and experimental thicknesses is. In-deed, Fig. 5b shows that the relative deviation, defined here by thedifference between df and h0 divided by the last parameter, growsas the thicknesses decrease. For volumetric flow rates higher than

Fig. 6. Partial direct visualisations of film surface for different flow rates withdimensions of roughly 40 � 80 mm2. (a) d�1. (b) d�2. (c) d�3.

Fig. 7. v as a function of standard film thickness deviations.

Fig. 8. v as a function of the Takeshi [10] parameter d�.

Fig. 9. Time-averaged temperature given by LCI TLCI in comparison to the oneobtained by thermocouple TTC – (1) corresponds to d�1, (2) to d�2 and (3) to d�3. Thedistance between the two dashed lines corresponds to a total range of 4 �C (±2 �C oneach side of the mean line).

518 N. Borgetto et al. / Experimental Thermal and Fluid Science 44 (2013) 512–519

10 mL min�1, the relative difference between theoretical andexperimental data remains lower than 10%. In this context, it canbe concluded that the Nusselt theory correctly predicts the time-averaged thickness of a liquid film injected through the porousmedium before flowing on the vertical wall.

In our experiments, ReT varies from 1 to 180, and We is between50 and 1575. Among the flow rates studied, three were selected inthis paper to illustrate qualitatively the structure of the liquid/gasinterface. Fig. 6 shows direct visualisations of these three cases:d�1 < 1 (a), d�2 � 7 (b) and d�3 � 20 (c). Also represented in the pic-tures (by +) is the position of the zone studied by LCI. These figureshighlight flow structures that are consistent with the previouswork of Takeshi [10]. Indeed, the film surface remains flat inFig. 6a and, as parameter d� increases, waves are formed (b) untila ‘‘chaotic’’ appearance of the flow emerges (c).

These behaviors have an influence on the values of the valida-tion rates of measures, v as given by Eq. (9). Fig. 7 shows the val-idation rate v as a function of the standard film thicknessdeviation, rdf. At low rdf values, v remains close to one and de-creases suddenly when the standard film thickness deviation ap-proaches unity. Accordingly, a liquid film cannot be consideredflat when its standard thickness deviation is higher than 1 lm.

It is thus possible to distinguish ‘‘flat’’/wavy films by their vali-dation rate v. In Fig. 8, we have plotted parameter v as a functionof d� for several stationary (ambient) temperature conditions, be-tween 15 and 20 �C. The validation rate decreases from approxi-mately 100% down to 25% when d� approaches 1. This trend isconsistent with several numerical results from literature [10].

These different regimes and wall temperatures are coupled withheat transfer phenomena inside the film. Indeed, data were ac-quired for several prescribed differences in temperature betweenthe wall and the liquid inlet (Tin � Tw) and for several film thick-nesses and flow regimes. Three similar configurations, d�1 (1), d�2(2) and d�3 (3), corresponding to the liquid/gas interfaces as shownin Fig. 6, were studied. The results are summarized in Fig. 9. Thetime-averaged temperature given by LCI TLCI is compared withthermocouple measurements TTC for these three configurations.For the first two (1, 2), both techniques provide very similar values

N. Borgetto et al. / Experimental Thermal and Fluid Science 44 (2013) 512–519 519

(TLCI � TTC), and absolute deviations between the two methods re-main lower than 2 �C. In the third case (3), which corresponds tod�3 � 20 and an average film thickness of 180 lm, a systematic dif-ference appears for the highest wall temperatures (30–40 �C).Fig. 6c shows that, in this configuration, the zone where measure-ments were taken is the seat of the waves at the free surface of thefilm. In addition, both Tin � Tw and the mean film thickness arefound to be higher in this case. These thermal gradients explainthe drift, which is not therefore connected to LCI measurementerrors.

Results demonstrate the feasibility of non intrusive temperaturemeasurements of interfaces between a vertical wall and a film flow-ing on it by LCI. They also highlight the intricacy of such measure-ments and their uncertainty estimations, which are closely relatedto the stability mechanisms at the free surface of the liquid film.

6. Conclusion

An academic experimental device was built with the aim ofanswering several questions related to the physical phenomenainvolved when a liquid fuel film flows on a vertical wall and inter-acts with a flame. In order to study the film properties, theLow-Coherence Interferometry (LCI) technique was developed. Ina simple, non reactive configuration (wall/liquid film/gas), we wereable to: (1) measure local and instantaneous liquid film thicknessand (2) estimate the temperature of the wall over which the liquidfilm is deposited. This enabled us to observe some of the advanta-ges and limitations of the LCI technique which was, to our knowl-edge, being used for the first time in such a context. Regardingliquid film thickness, good concordance was reported between atheoretical model, given by the Nusselt solution, and the experi-mental data. The main sources of error were found to be relatedto the volumetric flow rate measurements required to apply thetheoretical model. In addition, a good concordance, in terms ofphysical behavior, was observed with numerical predictions fromTakeshi [10] related to the development of surface waves. LCIwas found to be suitable for measuring thin liquid film thicknesseswith a good degree of accuracy in the presence of high thermal gra-dients and reactive flows [6]. However, some limitations were alsopointed out: (1) the minimum thickness measurement (which isfixed by the coherence length of the light source) is roughly20 lm in our case; (2) surface waves on the liquid film affect mea-surements which may be a problem for the characterization of non

flat films; (3) a dynamic description of liquid film morphologyrequires high interferogram sampling frequencies. In order toachieve these, the MM translation stage needs to be driven alonga reduced interval that decreases the maximum film thicknessmeasurement. The MM translation stage displacement frequencycan also provoke a vibration problem which deteriorates measure-ment accuracy. The feasibility of the simultaneous wall tempera-ture measurement was shown. This is another advantage of thisoptical technique, but further work is required in order to deter-mine precisely the associated uncertainties.

Acknowledgement

The project was supported by the National Agency of Research,France, within the framework of the ACTING-CO2 program (ANR-09-VTT-02-02).

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