Ultrasonic measurement of condensate film thickness

Phase change involving liquid films is important to a variety of terrestrial and space applications. These include two-phase thermal control systems for environment control and life support, including humidity control, air revitalization, water purification, and heat rejection, as well as thermal management of spacecraft components, all of which depend upon condensation and evaporation.^1–4 Many more applications are dependent specifically upon evaporation, including fuel vaporization,⁵ flame propagation over liquid fuels,⁶ hardening of plastics, and numerous coating situations.⁷ Phase change processes in space are a concern because buoyancy effects are important in terrestrial applications but are nearly absent in phase change in microgravity. The physical phenomena, including convection, operative in condensing and evaporating films have been studied extensively under a variety of conditions (see, e.g., Som et al.,⁸ and references therein) but accurate, dynamic, nonintrusive measurements of film thickness are lacking.

Previous work involving ultrasound to investigate liquid films undergoing phase change separated the signal processing into two regimes.^9,10 Ultrasound signals corresponding to films where individual echoes overlapped were studied in the frequency domain, while thicker films were studied in time domain. As implemented, this technique was limited to studying films greater than $50μm$ where standing waves in the film produce strong signals at the transducer.

The current work applies normal-incidence ultrasonic probing with an array of transducers to the study of the behavior of liquid films undergoing phase change. Described is a modified time-of-flight signal processing technique that is able to measure dynamically the thickness of liquid films on a metal substrate over a broad range (from less than $1∕6$ of the acoustic wavelength, where significant ringing is present, to above several millimeters) with a precision of approximately $±1μm$ and accuracy better than $±10%$ in trial experiments.

The technique uses scripts and functions in MATLAB to create a simulated signal matched using a least-squares technique to each acquired signal. Matching is accomplished by varying the film thickness in the simulation, which varies the time delay of each echo in the reverberant three-medium problem. The simulated signal is created by first calculating the amplitude of each wave based upon its ray path in a characteristics diagram and the material properties. The wave form of each wave is the same as, or an inverted form of, the wave form measured in normal reflection from the metal/vapor interface. The individual waves with appropriate time delays are then summed to create the simulated signal. The technique has the benefit of being able to process films from as little as $8μm$ to more than $5mm$ thick, even if the individual waves overlap in time.

In the current experiment, the ultrasound transducer is separated from the fluid layer (the region of interest) by a copper plate approximately $3mm$ thick on which a condensing $n$-pentane film resides. An initial pulse is sent from the transducer, and when it encounters the change of acoustic impedance at either the metal/liquid interface or the liquid/vapor interface, one wave is transmitted and another wave is reflected.

The amplitudes of the reflected and transmitted waves relative to the initial wave are predicted by Eqs. (1) and (2), where $ρ$ and $c$ represent the material density and sound speed, respectively, and subscripts 1 and 2 indicate material properties before and after the interface, respectively,

A simulated signal is created as part of the postprocessing of each ultrasound signal captured during the experiment. The time between successive echoes is varied in the simulated signal in order to minimize the sum of the squares of the residuals between the simulated signal and the acquired signal. Knowing the time between successive echoes in the simulated signal thus yields the film thickness. The details on the signal simulation method and the film thickness measurement are presented in the following section.

The simulated signal is created by summing many individual waves corresponding to distinct echoes received at the transducer from the metal/liquid and liquid/vapor interfaces. An individual wave received at the transducer is identified as $wave(n,p)$⁠, where $n$ is the number of round-trip passes through the liquid layer and $p$ is the number of round-trip passes through the metal layer, and has a corresponding amplitude coefficient, $amplitude(n,p)$⁠, and a time delay based on $n$ times twice the fluid thickness plus $p$ times twice the metal thickness. Several unique paths have the same delay and sum to form a single wave, $wave(n,p)$⁠, as seen in a propagation diagram (Fig. 1). All signals from the paths making up one wave have the same delay but have different magnitudes and may be amplitude inverted from one another. The number of different paths, $N$⁠, as a function of $n$ and $p$ used in calculating the relative $amplitude(n,p)$ corresponding to $wave(n,p)$⁠, is given by

Figure 1 shows the paths for the wave where $n=2$⁠, $p=3$⁠, wave(2,3). In this case there are six paths, as detailed in Table I. The vector corresponding to the first path, [2 0 0] (see Table I), would indicate that the path includes two passes through the liquid layer in the first pass through the metal layer and zero passes through the next two passes through the metal layer. This path is represented as the red line in Fig. 1.

Given the path vector, the number of reflections and transmissions for each interface can be counted. Then, when coupled with Eqs. (1) and (2) and the known material properties, the amplitude for each path relative to the initial pulse amplitude can be calculated. The relative amplitude of the wave is the sum of the amplitudes of each path.

The following technique is used to account for the attenuation in the metal substrate material and to set the absolute amplitude and the wave form of the waves. The reflections are measured without the liquid film present. This is $base(n=0,p)$⁠. The base wave form consists of a few rapidly decaying cycles at $20MHz$⁠, the result of a very short electrical excitation of the resonant PZT source. After a correction to the base amplitude for the slight difference in $R$ [Eq. (1)] with and without the film present, $wave(0,p)$ is set equal to $base(0,p)$ and more generally, $wave(n,p)$ is defined by $base(0,p)×amplitude(n,p)∕amplitude(0,p)$⁠. The frequency dependence of the attenuation in the metal is implicitly taken into account in that the measurement sets the amplitude and the wave form after propagation through the metal. The unknown reflection coefficient at the transducer/brass interface is similarly accounted for. The attenuation in the liquid layer, which is much smaller than the loss in the metal, is neglected.

The final step is to factor in the arrival times of the waves and sum them into one simulated signal. In other words, the $wave(n,p;t)$⁠, where previously the time, $t$⁠, has been suppressed, is delayed to $wave(n,p;t-nΔtliquid−(p-1)Δtmetal)$⁠, where $Δtmetal$ is the time between successive waves from the metal/liquid interface (i.e., twice the known metal thickness divided by sound speed of the metal) and $Δtliquid$ is the time between successive echoes from the liquid/vapor interface (i.e., twice the unknown thickness of the liquid divided by sound speed of the liquid). This delay sets the first reflection to arrive at time zero. Then all the waves are summed over $p$ and $n$⁠.

The free parameter $Δtliquid$ is directly proportional to the film thickness and matching the summed simulated signal to a signal acquired during the experiment by varying $Δtliquid$ allows for measurement of the film thickness.

The ultrasound system includes a digital oscilloscope, a preamplifier, a multiplexing thickness gauge, a multiplexer, four transducers, and a desktop computer. The transducers have a $20MHz$ center frequency and a $3-mm-diam$ element. The oscilloscope acquires the signals at $4Gsamples∕s$⁠. The thickness gauge acts as a voltage spike source and a transmit/receive switch. The transducers are excited with a single high voltage spike. Twenty megahertz sources, the highest frequency readily available, were chosen and a single electrical impulse about $20ns$ in duration was used in order to maintain the shortest acoustic pulse possible, thus maximizing the chance of resolving two pulses. With the technique described here this was less important. The thickness of the copper plate was minimized in order to minimize attenuation, which allowed for the high, $20MHz$⁠, acoustic frequency.

The gauge is set to switch between the multiplexer channels $20times∕s$⁠; therefore, a wave form from each of the four transducers is captured at $5Hz$⁠. After switching channels, the gauge sends a pulse to the transducer. This pulse triggers the oscilloscope, which acquires the pulse and echo signals from the preamp. Once triggered, the oscilloscope digitizes and sends the wave form to the computer via a USB connection. The computer stores the wave form and instructs the scope to wait for the next trigger.

The experiment occurs in a pressure-controlled cylindrical test chamber approximately $12.5cm$ in diameter by $15cm$ tall. The base of the cylinder is the copper test block, the top is a glass viewing window, and the sides are Teflon™. $n$-pentane is supplied from a heated vapor generator, and the pressure is controlled using an electronic backpressure regulator. The more complicated problem of a downward-facing, unstable film was studied by Som et al.⁸ in a test chamber similar to the current work.

The system was initially tested in simple control experiments where a known volume of fluid was applied to the copper surface and, therefore, the thickness was approximately known for comparison with the ultrasound measurements. For brevity and because it is difficult to obtain very thin films with that technique, those experiments are not reported here. Here we report measurements compared to calculations of film thickness based on known condensation rates.

Once the copper plate has reached the experimental temperature, 100 wave forms are acquired for each transducer with no liquid present on the metal substrate. These are averaged for each channel and are windowed to form the base wave forms discussed earlier. The signals in these wave forms are the echoes from the metal/vapor interface [see Fig. 2(a)]. The length of the acquired signal is approximately five times larger than $Δtmetal$⁠.

During postprocessing, the base signals are shifted in time to set the center of the first metal/vapor echo at time equal to zero. As with the simulated signal, time signifies the time elapsed since the first echo from the metal/liquid interface. The temperature of the block does not change significantly (less than $0.5°C$⁠) so that these offset times do not change significantly during the experiment.

The waves from the metal/liquid interface are removed from the experiment wave form by subtracting out the base signal, Fig. 2(a), from the measurement with a film, Fig. 2(b), to produce the signal in Fig. 2(c). Note the $y$-axis changes because the waves from the liquid/vapor interface are much smaller than those from the metal/liquid interface. Next, a least-squares fit between the simulated and measured signal is performed. The sum of the squares of the residuals between the measured and simulated signals is minimized using $Δtliquid$ as a parameter. Figure 2(d) shows a simulated signal matched to an acquired signal. The function finds the best local fit, so some care must be taken when choosing the start point. If the explained variation, i.e., quality of fit or $R2$⁠, is unacceptably low the user is prompted to enter a new set of initial parameters or accept the current fit. If the revised parameters fail to achieve an acceptable solution the measurement is rejected.

The measured thickness of $n$-pentane films on the copper surface was in excellent agreement with the predicted condensation rate (Fig. 3). With the metal surface facing upward, the pressure of the $n$-pentane vapor in the surrounding test chamber was raised quickly. The corresponding increase in saturation temperature resulted in the metal surface becoming subcooled, causing pentane vapor in the chamber to condense on the surface. This experiment was chosen for two reasons. First, the film is flat and parallel to the copper and therefore is reasonably easy to measure with the ultrasound system. Second, it is possible to predict analytically the film thickness versus time. The heat flux per unit area at the film surface due to condensation is expressed as $q″=δtρhfg$⁠; where $δt$ is the derivative of thickness with respect to time, $ρ$ is the density, and $hfg$ is the latent heat of fusion. Heat flux per unit area conducted through the film is expressed as $q″=−k(∂T∕∂x)≈−k(Tsub∕δ)$⁠; where $k$ is the thermal conductivity of the fluid, $Tsub$ is the degree of subcooling, and $δ$ is the film thickness. Equating these expressions and solving for $δ$ yields Eq. (4), the established prediction for film thickness as a function of fluid parameters and time,

The fluid properties are taken from the NIST Chemistry Webbook.¹¹ The stated uncertainty in $k$⁠, $ρ$⁠, and $hfg$ are 1.0%, 0.2%, and 1.0%, respectively. The uncertainty in $Tsub$ is $0.1°C$⁠, yielding a total uncertainty in the thickness calculation of approximately 3.5%.

In Fig. 3, the ultrasonically measured film thickness versus time is compared to the thickness calculated with Eq. (4). The measured values are similar for all four channels and agree with the predicted values to within 10% [Fig. 3(b)] suggesting that the technique is accurate to at least this degree. Deviations appear around $15s$ into the test interval when the film is $150μm$ thick and continue through to the end of the experiment. These deviations are likely due to the copper plate not being perfectly level to the ground resulting in fluid motion within the film. The level was off horizontal by less than $18arcsec$⁠. The limitation in the accuracy appears to have been more a result of the difficulties associated with creating a perfect experimental configuration, specifically the lack of perfectly level test surface, than of the ultrasound technique itself. The measured thicknesses were also compared with smooth curves calculated for each channel using a 50-point Savitzky–Golay filter. That comparison indicated a deviation from the corresponding smoothed curves of less than $5μm$ [Fig. 3(c)] with a variance of approximately $1μm$⁠, suggesting that the precision of the measurements is approximately $1μm$⁠.

The current work describes a technique for film thickness measurement that allows for processing ultrasound signals where echoes from the distal layer, the layer of interest, may overlap with echoes from the proximal layer and/or other echoes from the distal layer. The same technique can also handle signals with thicker distal layers where echoes from the distal layer are received after the second echo from the proximal layer. Although shear waves and possible oblique reflections from the film surface were likely present and were neglected, the system proved to be precise to within $1μm$ and accurate to within 10%, with significantly greater accuracy likely possible with a more precisely leveled substrate material. The technique benefits from the requirement of carefully controlled temperature to conduct the condensation experiments, and thus, changes in temperature and sound speed are negligible.

In this work, the distal layer is an $n$-pentane liquid layer undergoing phase change. The proximal layer is a $3-mm$-thick copper plate. Using the technique described it is possible to match simulated signals to acquired signals for layer thickness ranging from $8μm$ to above $5mm$⁠.

This research was supported by NASA Cooperative Agreement No. NNC04GA76G under the technical management of E.L. Golliher of NASA’s Office of Biological and Physical Research as well as NSF Grant No. CBET-0651755. The authors thank Adam Maxwell, Brian MacConaghy, Jarred Swalwell, and Oleg Sapozhnikov of the UW Applied Physics Laboratory for their help in conducting preliminary experiments, and would also like to acknowledge many helpful suggestions from the Editor and reviewers.