Introduction

Thin falling films are widely used to enhance heat and mass transfer in process engineering, e.g. in cryogenic components, in wet cooling towers, in thermosyphon reboilers, in reactors and falling film evaporators. Thin film cooling of electrical components is a novel application. The heat transfer from a heating surface to a falling film is a very complex process with strong interactions between the wavy film, the pseudo stationary heat transfer and the mass transfer at the phase boundary. The local instantaneous thickness of the wavy films is an important property to characterize the falling film and the heat transfer phenomena.

Alekseenko et al. (1994) outlined the main techniques used in liquid film thickness measurement, such as the contact or needle-contact method (e.g. Brauer 1956; Ishigai et al. 1972), the radioactive additive method (e.g. Jackson 1955), the light absorption method (e.g. Staintrop and Allen 1965; Portalski 1963), the capacity method (e.g. Dukler and Berglin 1952), the electric conductivity method (e.g. Pokusaev et al. 1978), the shadow method (Kapitza and Kapitza 1965; Nakoryakov et al. 1976; Alekseenko et al. 1985), the fluorescence method (e.g. Hewitt et al. 1964; Hiby 1968), the light scattering method (e.g. Salazar and Marschall 1978) and the interferometric method (e.g. Unterberg and Edwards 1961).

Statistical characteristics of a falling film within a vertical pipe have been experimentally evaluated by means of the “parallel-wire conductance probe” technique, Karapantsios et al. (1989). The technique is based on the inverse proportionality between the electrical resistance and the liquid layer thickness covering the wires.

Lyu and Mudawar (1991) have developed an alternative technique based on the principle of hot-wire anemometry. The probe was made of a 0.0254 mm diameter platinum–rhodium wire having an estimated spatial and time resolution of 0.05 mm and 0.14 ms, respectively. The Reynolds numbers of their falling films ranged from 750 to 2925. They showed that the correlation between film thickness and liquid temperature is stronger at higher heat fluxes.

Nosoko et al. (1995) have used a needle contact technique to measure the phase velocity, the wavelength and the wavy peak thickness of two dimensional water films. Intermittent contacts of the film with the needle were detected using a laser beam. The thickness of the large wavy peaks could be determined from the known distance between the needle tip and the glass plate.

Leuthner et al. (1998) presented a high frequency impedance needle probe, which was totally immersed in the liquid film but had no contact with the wall. This probe was a modified version of a probe, which was already applied successfully in other two-phase flow studies (Auracher and Marroquin 1991). The technique was tested with a water film flowing down the outer wall of a vertical circular tube with both, isothermal and evaporating conditions.

An automated broadband capacitive thickness meter for cryogenic liquid films on free vertical surfaces and in narrow channels was described by Krotov et al. (1997). The capacitive probe provides the local wave profile with high time resolution (∼1 ms) and high spatial resolution (measuring spot ∼1 mm, perpendicular to the flow ∼10 μm). With an improved sensor Pavlenko et al. (2001, 2002) examined the falling laminar wavy film flow with intensive evaporation. The change of the probability density of film thickness, the phase velocity of the large waves as a function of the heat flux density was displayed within the range of Reynolds numbers from 32 to 103.

Capacitance probes were even used by Ambrosini et al. (2002) in order to investigate the statistical characteristics of the surface of a water film, falling down a vertical or inclined flat plate. The probes had an inner electrode with a diameter of 2.5 mm. Because of the size of the sensor, the film thickness was averaged and the high frequencies of surface fluctuations were damped.

Takamasa and Hazuku (2000) described a “laser focus displacement meter (LFD)” technique based on a confocal imaging principle. The conical laser beam emitted from a semiconductor laser passes through a half-reflection mirror and an objective lens, and then reaches the target surface. The scattering light from the target passing backwards through the objective lens is reflected by the half-reflection mirror and reaches a pinhole. The objective lens is moved at high frequencies by a tuning fork. The displacement of the fork, which is moved coincidentally with the lens is detected by a sensitive position detector. The reflection beam appears on a light-sensitive element at the rear of the pinhole, when the laser beam is focused on the target. The displacement of the target (i.e. the falling film surface) can then be determined by the displacement of the object lens when the signal from the sensing element is detected. The temporal resolution of the presented LFD system is about 700 Hz. The diameter of the beam spot on the target is 2 mm and the spatial resolution is 0.2 mm. The film thicknesses in the paper have shown unrealistic jumps, which are probably due to a problem with moving mechanical parts.

In order to investigate the hydrodynamics of three-dimensional waves in laminar falling films Adomeit (1996) and Adomeit and Renz (2000) used a fluorescence technique on the base of the work of Hiby (1968). A small amount of a sodium-fluorescine (C20H10Na2O5) was added to the film as a fluorescent tracer. The fluorescent light emitted by the fluid film was recorded by a CCD camera (50 Hz) equipped with a UV-filter. Its intensity is a function of film thickness, which needs to be calibrated. The resolution of this field measurement technique was about 0.02 mm. Moran et al. (2002) used photochromic dye (1′,3′,3′-trimethylindoline-2-spiro-2-benzospyran-TNSB) dissolved in the silicone fluid and activated by an UV nitrogen gas laser (wavelength 335 nm) to measure velocity profiles in the film flow. The velocity profiles and the film thickness were captured with a high speed CCD video camera at a rate of 186 fps.

Zaitsev et al. (2003, 2004) introduced a new fiber optical probe for measuring the film thickness. Schagen and Modigell (2005) developed a non-intrusive measuring procedure, which uses the optical characteristics of the indicator Diacetyl in a water film. The method allows the simultaneous evaluation of the film thickness from the fluorescence signal and the oxygen concentration distribution in the water from the phosphorescence signal.

The methods described above are often not accurate enough, some of them disturb the film flow and the calibration procedures are very complicated.

In this paper, two new techniques are presented and compared to measure the thickness and the wave velocity of transparent falling films. The first one is an extended method of the work by Cohen-Sabban (2001) the so-called “chromatic confocal imaging method”, who used this technique to scan the roughness of a solid surface. This chromatic confocal imaging method has been extended by the authors to measure the thickness of a transparent fluid layer. The other one is a modified fluorescence intensity method based on the work of Hiby (1968), Adomeit and Renz (2000). This modified method allows not only measurements of the instantaneous film thickness with high temporal resolution (Al-Sibai et al. 2002a; Leefken et al. 2004) but simultaneously of the wave velocity of a falling liquid film.

Description of the setup and the measurement methods

Experimental setup

The closed-loop test facility is shown in Fig. 1. A silicone fluid film flows down a Perspex plate (length 1600 mm, width 240 mm). The adjustable gap in the liquid distributor allows controlling the initial film thickness. At the end of the test section the film flows into a reservoir. The fluid is circulated by an adjustable flow-rate piston pump. The flow rate is measured by a positive displacement flow meter. The mean temperature of the film is measured with a thermocouple immersed in the reservoir. Silicone fluids (Polydimethylsiloxane, CAS No.: 63148-62-9, DMS-T01.5, T05 and T12, see Table 1) with different viscosities are used as test fluids. The measurement systems are installed 1200 mm downstream from the film-flow inlet. Experiments have been carried out at atmospheric pressure and ambient temperature.

Fig. 1
figure 1

Experimental setup

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Table 1 Physical properties for silicone fluids at T=25°C
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Silicone fluids were selected as working fluids for different reasons. Since they possess a small surface tension compared to water, film flows without streak like structures can be generated along the 240 mm planar width of the test section. A further reason is that silicone fluids are available with a large range of viscosities. Thus, with the applied silicone oils, a wide spectrum of Reynolds numbers \((Re = \ifmmode\expandafter\dot\else\expandafter\.\fi{{\text{m}}} \;\eta^{{- 1}} b^{{- 1}}) \) and Kapitza numbers \((Ka = \sigma ^{3} \rho ^{{- 3}} \gamma ^{{- 1}} \nu ^{{- 4}} )\) can be achieved. The physical properties of the silicone fluids at ambient temperature (25°C) are listed in Table 1.

A range of Re=10–700 for T01.5, Re=10–450 for T05 and Re=2–85 for T12 at various inclination angles has been examined.

The chromatic confocal imaging method

The principle of the chromatic confocal imaging method and the scheme of the CHR 450 unit, developed by STIL Corp., Cohen-Sabban (2001), are presented in Fig. 2. This measuring technique is normally applied to measure the surface structures of solid bodies. The method uses the confocal imaging principle and the effect of the chromatic aberration of a lens.

Fig. 2
figure 2

Principle of the chromatic confocal imaging method

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The light of a polychromatic point source (halogen lamp, 50 W) passes through a fiber-optic link with a fiber-optic coupler, which works as a half-transparent mirror. The lens splits the white light from the polychromatic point source into a continuum of monochromatic images with focal points at several distances, due to the chromatic aberration. The light is reflected from the measuring surface. Predominately, the part of the spectrum, which is in focus, is conducted to the spectrometer by a fiber-optic link and a spatial filter. A spectrometer measures the intensity of the light spectrum. The analogue signal from the spectrometer is converted into a digital signal. The appropriate distribution function and the wavelength at the maximal intensity, which lies in the focus, are evaluated by an optimization method. This wavelength is converted into a position within the view field of the optical sensor by means of a calibration curve. The converted position data are passed on to the rapid COM port (RS 232, 460 800 Baud rate). The sampling rate is adjustable between 0.4, 1, 2 and 4 kHz depending on the reflection property of the measured surface. For too large sampling rates, and as a consequence, the intensity maximum may be below the noise level for small integration times. Too large integration times result in a data overflow of the spectrometer.

In this paper, the chromatic confocal imaging method has been extended to measure unsteady falling film thicknesses and statistical characteristics of laminar wavy films. The extended technique allows evaluating very accurately the instantaneous and time resolved film thicknesses.

Most of the experiments were carried out with a frequency of 2 kHz. The relation between the detected wavelength with the maximal intensity and the position of the view field of the optical sensor is provided by the manufacturer. The calibration procedure is described by Lel et al. (2004). The error of the CHR 450 unit is about 1 μm for solid surface measurements, and the diameter of the measuring spot is 10 μm.

If a transparent film is located within the field of view of the optical sensor, both, the wall and the film surface produce a signal with two peaks at λW and λS, respectively (see Fig. 3).

Fig. 3
figure 3

Spectrum of the light reflected at the film and the wall surface

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The real film thickness δr can be evaluated from the optically measured one δopt using the optical law:

$$\delta _{{\text{r}}} \cong n\delta _{{\text{opt}}} ,$$
(1)

where n is the refractive index of the fluid and δopt is the corresponding distance between the positions of the two peaks λW and λS within the view field of the optical sensor assuming air conditions (see Fig. 4a). This function is a standard feature of the CHR 450 unit.

Fig. 4
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Light beam path through a film (a) film without waves, (b) wavy film

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Because, in the present case, the intensity of the wall signal is about ten times stronger than the weak signal of the film surface, which is often within the noise level, the chromatic confocal imaging method had to be extended. The new film thickness measuring method refers only to the signal reflected from the wall. First, a reference zero position of the wall without the film is recorded. Then the film is applied and the signal from the wall is measured again. The difference between both signals δW is a measure of the film thickness. The real film thickness δr can be found from the following relation (see Fig. 4a):

$$\delta _{{\text{r}}} = \delta _{{{\text{opt}}}} + \delta _{{\text{w}}} = \frac{{\delta _{{\text{r}}}}}{n} + \delta _{{\text{w}}} $$
(2)

or

$$\delta _{{\text{r}}} = \frac{{\delta _{{\text{w}}} n}}{{(n - 1)}}$$
(3)

If the film is very thin, the two peaks are close to each other and the signal from the film surface influences the wall signal. The minimal film thickness, which can be measured is in the range of 200 μm.

If waves occur, the film surface is no longer parallel to the wall and the use of the Eq. 3 is incorrect, except at the local minima and maxima of the waves. In general, the measured film thickness data are slightly too high (Fig. 4b). But it can be shown that in most film flows the amplitude of the waves is small compared to the wavelength and the measuring error due to film curvature is only about 3 μm. For acquisition, post processing and data analysis a software based on “LabVIEW” has been developed.

From the results shown in Fig. 7 it can be seen that at the backside of large waves and in the residual layer area, where the change of the film thickness is small, the technique works perfectly and film thickness profiles can be evaluated with very high temporal and spatial resolution. However, at steep wave fronts, the amount of reflected light collected by the spectrometer is too low to be evaluated accurately. These data are invalid and a linear interpolation is used instead.

The fluorescence intensity method

With the modified fluorescence intensity method (Al-Sibai et al. 2002b; Leefken et al. 2004) the thickness and the wave velocity of a transparent film can be determined from the intensity of the excited fluorescence light. The local intensity of the fluorescence light depends, on the one hand on the local thickness of the fluid layer and on the other hand on the intensity of the excitation as well as on different geometrical parameters like the angle of incidence and the viewing angle. In the presented experiments, soluble fluorescence dye (Cumarin 152a) is added to the film fluid, since the fluorescence of the pure silicone fluid is too small. The excitation is caused by a laser diode with a wavelength of 405 nm. The nonlinear Lambert–Beer law is used to taken into account the influence of the molecular concentration of the fluorescence indicator in the film fluid and the extinction coefficient. A calibration procedure is used to determine this relation.

A uniform temperature level for calibration and measurement is important due to the temperature-dependence of the concentration and the extinction coefficient. In the case of the wavy film flow only small errors are to be expected due to reflections at the flat-wave flanks. The principle of the measuring technique is shown in Fig. 5.

Fig. 5
figure 5

Principle of the fluorescence intensity technique

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A diode laser with a power of 2.5 W produces a laser beam with infinite focal length and a wavelength of 405 nm. After a lens combination, the beam diameter is reduced and moves parallel through a half-wave plate over a mirror into a beam splitter crystal. In this crystal, the polarized light is split up into a perpendicular and into a parallel part. One polarization direction is led directly through the crystal, the other one goes through the crystal and appears parallel at a distance of 2.7 mm from the other beam. The two beams are reflected by a dichromatic mirror and led to the falling film. The fluorescence particles fluoresce in a wavelength range of approximately 400–570 nm.

The measuring spots with a diameter of 0.33 mm are magnified by a factor of three through a biconvex lens and led to the ends of two plastic fibers with a diameter of 1 mm. The fluorescence light passes the dichromatic mirror. The transmission of the filter is almost zero in the wavelength range of λ=400–430 nm and it is approximately 0.9 in the range from λ=460 to 560 nm. The two fibers lead the filtered fluorescence light over a second filter (yellow filter GG 475, edge filter with λ=475 nm) to the two photomultipliers. The photomultipliers convert the intensity of the fluorescence light into a voltage signal (0–10 V). The photomultiplier and the data recording allow scanning rates of up to 50 kHz.

A further important property of the film flow is the wave velocity because it extends the film thickness measurements from the time axis to the length information. Hence the wavelength and its distribution can be determined. Since the distance of the two measuring positions is known (Δ =2.7 mm) and the temporal distance of the two measuring signals can be found by cross correlating the signals, the wave velocity of the film-flow can be determined. Figure 6 shows the raw data of the time course of the fluorescence signals for the two measurement positions (signals 1 and 2) and the computed temporal distance Δt=0.009 s.

Fig. 6
figure 6

Raw data of the film thickness profiles of the two channels

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Because the signals differ in magnitude due to the adjustment of the optics, a specific calibration curve for each measurement position is used to evaluate the film thickness due to the accuracy of the calibration the error of the film thickness is in the range of ±20–50 μm.

Experimental results

Simultaneous measurements of the film thickness with the CHR-System and of wave velocities with the fluorescence intensity method were carried out at different inclination angles (φ=13, 30, 60 and 90°) and with silicone fluids DMS-T01.5, DMS-T05 and DMS-T12. The measuring position was 1135 mm from the film inlet.

Comparison of measuring systems

In order to show the capabilities of the two methods, film thickness measurements for a silicone fluid with Re=17 are compared in Fig. 7.

Fig. 7
figure 7

Comparison of measured film thicknesses for Re=17

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One can see that the fluorescence-intensity method allows scanning rates of about 50 kHz, and the film contour can be represented very accurately. On the one hand the confocal imaging method allows scanning rates only up to 4 kHz. At the wave crest, data must be interpolated; on the other hand the chromatic confocal imaging method needs no calibration and the accuracy is in the range of 15 μm. Another advantage of the latter method is the simpler measuring setup, because the distance between the measuring pen and the film can be in the range of a few centimeters. According to the focal length of the system, the spacing between the fluorescence intensity system and the film surface must be adjusted accurately in a range of a few millimeters.

Film thickness

Figure 8 shows typical films for different Reynolds numbers and angles of inclination measured by the chromatic confocal imaging technique. It is well known that a falling film at low Reynolds numbers consists of a residual layer with a constant thickness and large waves. Large waves interact with each other very rarely. By increasing the Reynolds number, both frequency and amplitude of the large waves grow interacting more frequently and surface capillary waves emerge.

Fig. 8
figure 8

Typical film thickness profiles for different Reynolds numbers and inclination angles

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Since it is well known that at low Reynolds numbers, the film’s average thickness is well described by Nusselt’s film theory, the measured data were averaged and compared to Nusselt’s theoretical result which is given by Eq. 4.

$$\delta _{{\text{Nu}}} = \sqrt[3]{{\frac{{3v^{2} \operatorname{Re}}}{{g\,\sin (\varphi )}}}}$$
(4)

As can be seen in Fig. 9 the experimental data agree well with theory for low Reynolds numbers.

Fig. 9
figure 9

Average film thickness measured by the chromatic confocal imaging technique for different inclination angles and comparison with the Nusselt’s theory for laminar films. a φ=13°; b φ=30°; c φ=60°; d φ=90°

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In the laminar region there is a good agreement between Nusselt’s theory and the presented data. In the transient and turbulent flow region, the measured values tend to be larger for T05 at 60 and 90°, as well as for T01.5 at 13° and slightly smaller for T12 at 60° than Nusselt’s prediction for laminar films.

To compare the present data with literature results the data are presented in Fig. 10 in terms of a dimensionless mean film thickness defined by

$$\delta ^{*}_{{\text{m}}} = \delta _{{\text{m}}} (g\,\sin \,\varphi /v^{2} )^{1/3} . $$
(5)

The own results are in good agreement with data in the transient and turbulent flow region presented in the literature by Brauer (1956), Karapantsios and Karabelas (1995), Takamasa and Hazuku (2000) and Ambrosini et al. (2002).

Fig. 10
figure 10

Comparison of the measured dimensionless mean film thickness with results from the literature and with Nusselt’s theory for laminar films

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From the present data for the silicone fluids DMS-T01.5, DMS-T05 and DMS-T12 measured at the inclination angles φ=13, 30, 60 and 90° the following correlation for the mean film thickness can be found

$$\delta ^{*}_{{{\text{m},}1}} = 1 + 0.615\operatorname{Re} ^{{0.47}} \quad {\text{for}} 2 < \operatorname{Re} < 700.$$
(6)

Wave velocity

Figure 11 shows the results of the evaluation of the mean wave velocity for the silicone fluids DMS-T01.5, DMS-T05 and DMS-T12 measured at the different inclination angles φ=13, 30, 60, and 90°. The results are presented in terms of the dimensionless wave velocity U * w:

$$U^{*}_{\text{w}} = \frac{{U_{{\text{w}}}}}{{(vg\,\sin\,\varphi)^{{1/3}}}}.$$
(7)
Fig. 11
figure 11

Dimensionless mean wave velocityU * w as a function of the film Reynolds number for different silicone fluids (T01.5, T05 and T12) and inclination angles (φ=13, 30, 60 and 90°). The standard deviation is σ=0,05 m/s

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The measured mean wave velocity is satisfactorily correlated (Al-Sibai 2005) as

$$U^{*}_{{\text{w}}} = 5.5\operatorname{Re} ^{{0.36}} \quad {\text{for}}\;3 < \operatorname{Re} < 700.$$
(8)

The maximum deviation of the above correlation and the dimensionless wave velocity obtained from experimental data is less than ±20%.

In Fig. 12, this correlation is compared to some experimental results form Ambrosini et al. (2002), Brauer (1956), Jones and Whitaker (1966), Takamasa and Hazuku (2000) and Stainthorp and Allen (1965). However, only data for vertically falling water films could be found.

Fig. 12
figure 12

Comparison of experimentally determined wave velocities of different authors for vertically falling films

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For Reynolds numbers Re<150, the experimental results deviate less the ±10% from the correlation. In the range 150< Re<1500, the deviation is in the range of +35%.

Summary

In this work, two different methods for measuring the thickness of wavy films are presented. The first method, the chromatic confocal imaging technique was extended for wavy transparent films. Because the signal of the film surface was too weak and could not be separated from the noise, only the intense signal of the wall, which is modified by the film was used. Because the film thickness changes only moderately the effect of the light reflection is small and measurements could be carried out in almost all areas of the waves. Critical situations with invalid data are found only in the front of large waves where the film thickness is linearly interpolated.

The accuracy of the local film thickness measurements depends on the optical components and on the accuracy of the refractive index. It was estimated that the total accuracy of the measured film thickness is in the range of 15 μm, whereas the error due to the optical components in the present configuration is about 1 μm. The diameter of the measuring spot is 10 μm. The time resolution depends on the optical properties of the solid wall and is in the range of 1–4 kHz.

The second method for measuring film thickness and wave velocity simultaneously is a fluorescence intensity technique. For the measurement of the wave velocity, the original film thickness measuring system was modified and extended by introducing an additional measuring spot.

A disadvantage of the fluorescence intensity technique is the need of a calibration and the limited spatial resolution in comparison to the chromatic confocal imaging technique. Regarding the time resolution the fluorescence intensity technique has a much higher time resolution compared to the confocal imaging technique. A combination of both methods showed the best results.

Film thicknesses and wave velocities were presented in a Reynolds number range from 2 to 700 for silicone fluids with different viscosities and various inclination angles. The experimental data of average film thickness agreed well with published experimental and theoretical studies for wavy films. The measuring systems are currently applied to investigate the influence of the wave characteristics to the heat transfer in falling liquid films.