- Email: [email protected]
Determination of microchannels geometric parameters using micro-PIV
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Determination of microchannels geometric parameters using micro-PIV Gonc¸alo Silva, Nuno Leal, Viriato Semiao ∗ Mechanical Engineering Department, Instituto Superior Tecnico, Universidade Tecnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
a b s t r a c t The precise characterization of microgeometries is a crucial task in the study of flow phenomena at these scales. Since the use of conventional optical microscopes is somehow limited in terms of accuracy, the present work studies the use of micro-PIV measurements to characterize a rather irregular microchannel cross-section. The micro-PIV is employed with a spatial resolution of 23.68 m × 23.68 m allowing for the location of the microchannel walls with an accuracy, in average, 10 times higher than that provided by the use of a conventional optical microscope. The accuracy of the micro-PIV results was validated by comparing the volumetric flow rate yielded by the integration of the micro-PIV velocity profiles against that supplied by the syringe pump employed in this experimental work. The 3% difference revealed the good quality of the measurements, demonstrating the potential of the micro-PIV technique to characterize both the flow kinematic parameters and the enclosing geometry. © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Micro-PIV; Microchannels; Geometry definition; Microfluidics
1.
Introduction
The study of fluid dynamics at micrometric scale geometries still presents conflicting and controversial results. Despite being nowadays established that decreasing the length scale makes relevant the phenomena traditionally neglected in macroscale flow theory, it is proved that the incorrect estimation of the experimental uncertainties can lead to misleading reports of those relevant phenomena, Gad-el-Hak (1999), Hetsroni et al. (2005), Bayraktar and Pidugu (2006) and Kandlikar et al. (2006). A simple error propagation analysis shows that, no matter what parameter one intends to measure, the overall experimental error increases due to the greater influence of the uncertainty in the geometrical characterization of the fluid flow domain as the length scale decreases (see, e.g. Hetsroni et al., 2005). Moreover, due to uncertainties in the manufacturing process and, sometimes, due to the lack of information from the manufacturer, the geometry dimensions and shape are generally not known to the desired accuracy.
The influence of an inaccurate geometrical characterization in microfluidic flow studies is not a new issue. In fact, its influence has repeatedly been reported as one of the main causes for obtaining incorrect results in experimental microfluidic studies, Hetsroni et al. (2005) and Magueijo et al. (2006). Pressure drop measurements in fully developed flows and its comparison against the results predicted by the conventional Fluid Mechanics theory applied to macroscale flows is perhaps the most notorious example of a situation where the flow geometry has a major influence in the accuracy of the final results. Distinctive examples of this situation can be found in the open literature. For instance, Guo and Li (2002) used a 40× optical microscope to characterize the inner diameter of a circular glass microtube obtaining a value of 84.7 m. The inclusion of this value in the pressure drop computation yielded values considerably higher than those predicted by the conventional Hagen–Poiseuille theory. The same measurements conducted with both a 400× optical microscope and a scanning electron microscope (SEM)
Abbreviations: CCD, charge coupled device; NA, numerical aperture; PIV, particle image velocimetry; SEM, scanning electron microscope; SPM, scanning probe microscope. ∗ Corresponding author. Tel.: +351 218417726; fax: +351 218475545. E-mail address: [email protected] (V. Semiao). Received 22 February 2008; Received in revised form 27 August 2008; Accepted 27 August 2008 0263-8762/$ – see front matter © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2008.08.009
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
Nomenclature ddiff dp Dh e M n U V X Y zcorr Z
diameter of point spread function (m) particle diameter (m) microchannel hydraulic diameter (m) CCD pixel size (m) magnification refraction index fitted velocity profile (m/s) velocity vector (m/s) streamwise flow direction (m) cross-streamwise direction (width) (m) depth of correlation, half the thickness of the micro-PIV measurement volume (m) cross-streamwise direction (height) (m)
Greek symbols ız depth-of-field of a microscope objective (m) ε relative contribution of a particle displaced a distance z from the object plane wavelength of the emitted light (m)
revealed, for the same microtube, an inner diameter of 80 m, value that resulted in pressure drop values in good agreement with the theory. Another problem related to uncertainties in the geometrical characterization of the flow boundaries is described by Celata et al. (2006). These authors observed for a Teflon circular microtube with 304 m diameter the presence of slight discrepancies between the measured pressure drop and the value theoretically expected. The observation of SEM images of the microtube revealed deformations in the assumed cross-sectional circularity shape, i.e. instead of circular the microtube cross-section was best described by an ellipse. The comparison of the measured pressure drop data with the predicted Hagen–Poiseuille results, now for an elliptical cross-section, showed good matching. Based on the previous reasoning it is clear that optical microscopes and their conventional magnifications, usually below 100×, may not be particularly recommended if one needs to attain high accuracies. A related problem occurs when performing micro-PIV measurements. Since the micro-PIV technique is traditionally based on the use of optical microscopes, the visual determination of flow features is somehow limited by the accuracy of the optical system itself. To overcome this, avoiding the use of expensive equipment like the SEM or the scanning probe microscope (SPM), and achieving accuracies higher than those provided by a simple optical microscope, the micro-PIV velocity data can be used. This research work intends to perform a thorough study on the use of the micro-PIV technique to accurately characterize the microfluidic flow geometry, particularly its inner boundaries. The comparison of the results provided by micro-PIV measurements against those obtained through raw visual inspections with a conventional optical microscope is also performed in order to establish the validity of the former.
2.
Micro-PIV fundamentals
PIV is a non-intrusive optical technique used to measure the fluid motion. The motion of the fluid is determined by measuring the displacement of seeding particles in two recorded
299
images separated by a known period of time. Each image is divided into a certain number of cells, named interrogation areas, each of these containing a minimum amount of particles. For traditional cross-correlation algorithms, a minimum of five particles per interrogation area is usually required, whereas for more advanced correlation algorithms (e.g. ensemble time average algorithms) this requirement may not be necessary (Meinhart et al., 2000). The mean displacement of the amount of particles contained in each interrogation area is determined through a statistical correlation between the two images recorded. Micro-PIV is a technique that is based on the PIV concept. Yet, micro-PIV is applied to perform measurements in geometries with much smaller dimensions where much higher resolutions are required. Wereley et al. (2002) outlined three main aspects that differentiate micro-PIV from the traditional PIV: the flow field is volume illuminated instead of being illuminated by a laser light sheet, producing higher levels of background noise from out-of-focus particles if their density is too high; if particles are smaller than the laser light wavelength, which is needed in some studies where spatial resolutions of a few microns are required, light scattering techniques cannot be employed; Brownian motion of particles with diameters smaller than 1 m may compromise the evaluation of the flow velocity since that introduces an additional percentage of uncertainty in the final result. The first work where the PIV technique was successfully adapted for studying microscopic flows was carried out by Santiago et al. (1998). A pressure-driven Hele-Shaw with a cylindrical obstruction in the centre of a 120 m2 flow was studied and results were obtained with 3.45 m vectorto-vector distances with 6.9 m × 6.9 m spatial resolution. Later, Meinhart et al. (1999) measured the velocity field of a 30 m × 300 m glass channel with a spatial resolution of 0.9 m × 13.6 m in the wall normal and streamwise flow directions, respectively. Stone et al. (2002) used the micro-PIV technique applied to the study of flows in microchannels to determine the position of the microchannel walls within a spatial resolution of tens of nanometers for a single horizontal plane. The high accuracy of their measurements was obtained by using interrogation volumes of 0.9 m × 13.6 m × 1.8 m that already account for the existence of 50% overlap in the spanwise and streamwise directions. These authors were the first to propose the experimental technique discussed and developed in this work as an alternative to the application of specific microscopes that, as mentioned before, possess limitations for measurements in liquid filled microchannels. Such technique is applied herein at 61 horizontal measurement planes to reconstruct the entire microchannel cross-section.
3.
Experimental apparatus
In this work a microchannel manufactured by DantecDynamics® was used, which is shown in Fig. 1. This microchannel is manufactured by laser ablation in a polymeric chip (PMMA) and sealed with the same material. This ensemble allows for micro-PIV measurements with reduced optical aberrations. The study was performed in the rectilinear region of the microchannel, denoted in Fig. 1 by the zone enclosed by the elliptical line. The pressure required for the fluid to flow was provided by a syringe pump (NE-1000, New Era Pump Systems, Inc.® ). Error tests performed to this pump indicated the uncertainty
300
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
Table 1 – Maximum spatial resolution of commonly used air-immersion lenses
Fig. 1 – The studied microchannel integrated in a chip platform supplied by DantecDynamics® . of ±1.5 l/min. The fluid used in all experiments was deionised water. A micro-PIV system was used to study the fluid flow inside the above microchannel. A pulsed Nd:YAG laser New Wave® Solo II-15 ( = 532 nm) was used to illuminate the measurement section of the microchannel. Fluorescent nile red flow-tracing particles with 1 m of average diameter were added to deionised water in a proportion of 0.5 l (of a solution of 3.64 × 1010 particles/ml) to 1 ml of water. An epi-fluorescent filter was used to prevent the background noise having wavelength below the red light spectrum ( = 575 nm) to reach the Flowsense® 2M CCD (charge coupled device) camera. This camera has a 1600 × 1186 pixel2 resolution (pixel size = 7.4 m) with 8/10-bit intensity resolution. Time delays ranging from 100 s to 500 s between frames and a repetition rate of 15 Hz were used. In order to obtain magnified particle images the CCD camera was coupled to a Leica® DM ILM microscope with an air immersion lens of magnification M = 10 and numerical aperture of NA = 0.25. A schematic representation of the micro-PIV system used in this work is presented in Fig. 2.
4.
Measurement resolutions
4.1.
Optical microscope resolution
The images of the microchannel geometry are captured by a CCD camera connected to an optical microscope. Therefore, in a micro-PIV setup, the image properties are influenced by both the microscope and the CCD camera optical parameters. The maximum in-plane spatial resolution of any optical system is limited by light diffraction, a phenomenon consist-
M
NA
ddiff (m)
Maximum spatial resolution (m)
5 10 20 40 63 100
0.12 0.25 0.40 0.55 0.70 0.90
54.1 51.9 64.9 94.4 116.8 144.2
27.05 25.95 32.45 47.20 58.40 72.10
ing of an interference pattern that images the object as a central disk (the Airy disk) surrounded by concentric rings (the Airy rings), as for reported for instance by Inoué and Spring (1997) and Raffel et al. (1998). It can be demonstrated, using the so-called Fraunhofer approximation (Adrian and Yao, 1985), that the diameter of the Airy disk in the image plane, ddiff , is expressed by Eq. (1), where M is the lens magnification, is the wavelength of the emitted light and NA the lens numerical aperture: ddiff = 2.44 M
2 NA
(1)
The resolution of an optical system can thus be defined as 0.5ddiff , i.e. half the distance between two adjacent Airy disk centres. The definition of the spatial resolution by this criterion is known as the Rayleigh criterion (Inoué and Spring, 1997). Taking into account that the image in a CCD camera is formed by an array of pixels, the pixel size emerges as another limiting parameter to the maximum optical resolution since no geometry can be resolved below the pixel dimension. However, considering the pixel size of a CCD camera, which is usually of the order of 10 m or less, and the optical characteristics of commonly used air-immersion objectives, the maximum in-plane spatial resolution in a micro-PIV is limited by the light diffraction phenomenon. Table 1 illustrates several diffraction and resolution values for commonly used air-immersion lenses. Besides the maximum in-plane spatial resolution one also needs to take into consideration the out-of-plane resolution limit. The out-of-plane resolution is given by the depth-of-field of the optical system, ıZ , which can be viewed as an estimate of the distance that a microscope slide may be moved in the
Fig. 2 – Scheme of the DantecDynamics® assembled micro-PIV equipment.
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
4.2.
Table 2 – Depth-of-field of commonly used air-immersion lenses. M
NA
ız (m)
5 10 20 40 63 100
0.12 0.25 0.40 0.55 0.70 0.90
49.3 11.5 4.3 2.1 1.3 0.7
vertical direction while still maintaining focus on an infinitely thin specimen. According to Inoué and Spring (1997), the depth-of-field of a microscope objective is given by the sum of the focal depth due to diffraction, the first term on right hand side of Eq. (2), and the focal depth due to geometrical effects, the second term on right hand side of Eq. (2): ız =
n0 NA2
+
ne NA M
(2)
In the previous equation, n is the refractive index of the imaging medium, 0 is the wavelength of the imaged light in vacuum and e is the smallest distance that can be resolved by the image detector placed in the microscope’s image plane, which corresponds to the CCD pixel size in the present case. Table 2 illustrates the out-of-plane resolution achieved by commonly used air-immersion lenses with n = 1.00, 0 = 532 nm and e = 7.4 m.
301
Micro-PIV resolution
As previously stated, in the micro-PIV technique the image is divided into a certain number of cells named interrogation areas. To make full use of the information provided by the images, the interrogation areas are generally overlapped. As a result, and although each velocity vector represents the average velocity of the flow inside the interrogation area, the distance between neighbouring velocity vectors is greatly reduced due to the overlapping process. The distance between neighbouring velocity vectors is considered here to define the in-plane spatial resolution of the micro-PIV measurements. The velocity vector yielded by the micro-PIV technique characterizes the average motion of the flow or, more precisely, the average motion of the total amount of particles in that region within the interrogation area. Excluding the presence of severe velocity gradients, which may significantly deteriorate the correlation peak biasing the velocity measurement (Westerweel, 2008), as long as all particles stay within the interrogation area, the computed velocity vector is always an accurate measure of the mean flow velocity within the interrogation area. However, by construction, the micro-PIV technique always places the measured velocity vector in the centre of the interrogation area. This can lead to significant bias errors in the velocity representation, even in the case of minor velocity gradients. This error exists since one is assuming that the mean velocity is also located at the interrogation area centre. However, this assumption is only true when there is no velocity gradient or, when the velocity gradient exists but
Fig. 3 – Velocity profile within an interrogation area and corresponding histogram of velocities: (a) linear velocity profile, (b) histogram of velocities for a linear velocity profile, (c) parabolic velocity profile and (d) histogram of velocities for a parabolic velocity profile.
302
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
without compromising the overall results quality in terms of valid/rejected vectors, this spatial resolution was intentionally chosen to demonstrate that, provided the conditions previously discussed are verified, the method studied herein is applicable. Regarding the spatial resolution in the out-of-plane direction, since the flow field in micro-PIV is volume illuminated, this parameter is not just simply given by the lens depthof-field. This is so because there are particles that still emit enough light to contribute to the correlation signal despite being out-of-focus. Therefore, the resolution in the vertical direction, named by Olsen and Adrian (2000) as depth-ofcorrelation, is defined as twice the distance that a particle of diameter dp can be positioned from the object plane so that the intensity along the optical axis is an arbitrarily specified fraction of its focused intensity, ε. Beyond this distance, the particle’s intensity is sufficiently low and does not influence the velocity measurement. For this reason, the depth-ofcorrelation, 2Zcorr , can be viewed as the limiting out-of-plane spatial resolution of a micro-PIV measurement. According to the study of Olsen and Adrian (2000) the depth-of-correlation is given by the following equation:
it is constant, i.e. when the velocity changes linearly within the interrogation area, see Fig. 3a and the corresponding histogram, Fig. 3b. For non-linear velocity variations, like those sketched in Fig. 3c, the bias velocity error is present and is represented in Fig. 3d. Consequently, if one assumes that the presence of a velocity gradient does not lead to out-of-pair particles, the error produced by such gradient is only relevant when the interrogation area is so large that the inaccuracy resulting from the linear velocity assumption within the area is significant. Consequently, if the real velocity profile can be represented, without significant loss of accuracy, by piecewise linear functions (where the length of each section corresponds to the interrogation area length) the assumption of negligible velocity gradients is valid. This analysis leads to the conclusion that small interrogation areas provide the most accurate results since they minimize this error. However, considering the fact that the signal strength in the velocity computation process is proportional to the number of particle pairs within the interrogation area (Keane and Adrian, 1990; Westerweel, 1998), if one keeps the particles density constant it immediately follows that the interrogation areas size cannot be decreased below
2zcorr = 2
√ 1− ε √ ε
a certain dimension. For that reason, the interrogation area size must be optimally chosen so that the maximal signal strength is achieved with the less velocity gradient error contribution possible. A rule-of-thumb generally followed by the PIV community consists in choosing interrogation areas sizes that simultaneously ensures that, at least, five particle pairs are present (Keane and Adrian, 1990) and that the particles displacement does not exceed 1/4 of the interrogation area length (Keane and Adrian, 1990; Westerweel, 1998; Raffel et al., 1998). If the velocity gradient contribution can be neglected or, at least, if the velocity vector location within the interrogation area is not too inaccurate, the use of sub-pixel interpolation schemes (Willert and Gharib, 1991; Prasad et al., 1992) allows one to estimate the average velocity value inside the interrogation area to sub-pixel accuracy. In fact, as demonstrated by Prasad et al. (1992), as long as particles are resolved by 2–3 pixels, the accuracy in locating the maximum peak in the correlation plane, i.e. the precision in determining the velocity vector within the interrogation area is approximately 0.1 pixels. Therefore, even though the micro-PIV in-plane spatial resolution is usually of the order of several pixels, the kinematic information supplied by this technique, which is given by the mean velocity vector inside the interrogation area, is of the order of 0.1 pixels. If the information provided by the kinematic field can be somehow related to the determination of some spatial feature in the flow field, one will be able to spatially characterize that feature with accuracy higher than that provided by a conventional optical microscopy technique. Therefore, in order to achieve an in-plane spatial resolution of less than one pixel size, while ensuring that this resolution provides an accurate measure of the flow field, the presence of velocity gradients within the interrogation areas cannot be significant. In this study, and because the flow is fully developed, as described below, the in-plane spatial resolution was chosen to have a value of 23.68 m × 23.68 m, with 50% overlap, in a field of view of 1172 m × 887 m. Although the in-plane spatial resolution could be significantly decreased
2
dp 2 (n/NA) − 1 4
2
+
2
2 1/2
5.95(M + 1) 2 (n/NA) − 1 4M2
(3)
Since the depth-of-correlation value is always larger than the depth-of-field value of the used lens, the limiting outof-plane resolution of a micro-PIV measurement is always larger than that of a conventional microscope measurement. Table 3 illustrates common values of depth-of-correlations with air-immersion lenses assuming dp = 1 m and ε = 0.01 and absolute differences to corresponding depth-of-field values, z = zcorr − ız . As it can be seen in Table 3 the micro-PIV spatial resolutions in the out-of-plane direction are considerably larger than those provided by conventional microscopy measurements. This difference decreases significantly as the lens magnification and numerical aperture increase. To overcome this problem the scientific community has associated the micro-PIV technique with more advanced microscopy techniques. The use of laser scanning confocal microscopy (LSCM) is one of the most recognized examples. The use of such microscopes allows one to achieve high out-of-plane resolutions without compromising the in-plane spatial resolution, i.e. without having to use high magnifications. For instance, Kinoshita et al. (2005), using a micro-PIV system with a confocal microscope, performed velocity measurements in moving droplets inside microchannels with a field of view of 228 m × 171 m and a confocal depth of 1.8 m. The major drawbacks of using the confocal technique
Table 3 – Depth-of-correlation of commonly used air-immersion lenses assuming dp = 1 m and ε = 0.01 and absolute differences to corresponding depth-of-field values M
NA
2Zcorr (m)
5 10 20 40 63 100
0.12 0.25 0.40 0.55 0.70 0.90
161.9 34.2 12.8 6.5 3.7 1.5
z (m) ±31.65 ±5.60 ±2.10 ±1.15 ±0.55 ±0.05
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
are the considerably higher cost of the equipment required and the relatively lower state-of-development, when compared to the conventional micro-PIV technique. This fact is denoted by the limited range of applications that can use confocal micro-PIV with success at present (Lima et al., 2006). Despite of these drawbacks, the virtues of confocal micro-PIV are undeniable, particularly in situations where measurements with high out-of-plane resolutions are crucial. Similarly to the analysis performed for the in-plane spatial resolution, the out-of-plane resolution size, i.e. the thickness of the measurements control volume, can be considered adequate as long as the displacement of the particles in the out-of-plane direction does not exceed 1/4 of the control volume thickness and also that, within the interrogation area, severe velocity gradients in this out-of-plane direction do not occur. The lens used in this work has M = 10 and NA = 0.25, which corresponds to a depth-of-correlation value of 34.2 m (see Table 3). Therefore, in order to perform the reconstruction of the three dimensional velocity field with a vertical overlap approximately equal to the horizontal one, i.e. 50%, the horizontal planes were measured each 15 m in the vertical direction. This vertical motion is controlled by a stepper motor with reproducibility better than 1 m and a total precision of ±3 m that is controlled by a real time RISC processor. The stepper motor can be addressed with a resolution of 1.5 nm. The influence of the chosen in-plane and out-of-plane resolution values in the accurate description of the velocity field is performed below.
303
Fig. 5 – Measured velocity profile at X = 200 m of a horizontal measurement plane at Z = 500 m.
5. Estimation of the micro-PIV velocity uncertainty and validity of the flow fully developed condition The a priori estimation of the error associated with the velocity measurement with micro-PIV is a complex task since each micro-PIV realization comprises the participation of several distinctive parameters whose interaction, in practice, is almost impossible to quantify in detail (a deeper analysis of some of the most common PIV error sources can be found, for instance, in Raffel et al., 1998 or Wereley and Meinhart, 2004). Consequently, and in order to have an order of magnitude of the micro-PIV error present in the overall velocity measurements, an approach similar to one described in Silva et al. (2008) is adopted here. With this approach several measurement planes of the velocity field, as the one depicted in Fig. 4,
Fig. 4 – Example of an image of the studied microchannel at a horizontal plane Z = 200 m with measured velocity vectors.
Fig. 6 – Three-dimensional velocity profile built up from 61 two-dimensional horizontal velocity profiles, similar to that depicted in Fig. 5. are extracted along the microchannel height. In each plane Z = const, and for a constant streamwise location, X = 200 m in the present case, the velocity profile is computed. Fig. 5 shows an example of such velocity profile at Z = 500 m and X = 200 m. This procedure, applied to all the 61 horizontal measurement planes, allows for the reconstruction of the overall three-dimensional velocity profile as depicted in Fig. 6. The volumetric flow rate given by the volume enclosed by this three-dimensional velocity profile, which is computed through numerical integration, is compared against the corresponding flow rate supplied by the syringe pump, which is considered to provide a theoretical value. The difference between flow rates provides an indication of the bias error of the micro-PIV technique in the case under study. For the volumetric flow rate used herein, which is ca. 1 ml/min, the difference between the micro-PIV predicted value and the one supplied by the syringe pump, is ca. 3%. This indicates that, in average, the velocity information provided by the microPIV technique is expected to have an accuracy of ca. 3%. Since the micro-PIV data used in this analysis was obtained with the in-plane and out-of-plane resolutions described in Section 4.2, the accuracy of these resolutions is also denoted by
304
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
ca. 3% error. It should be stressed that the computation of the velocity field was performed with an ensemble time average correlation algorithm, as described by Meinhart et al. (2000) and Delnoij et al. (1999), in order to obtain a velocity field with the less unphysical velocity vectors. These unphysical vectors can be caused by the presence of both a low signalto-noise ratio in the correlation plane, caused by the required low seeding concentration, and the influence of the Brownian motion on the computation of the instantaneous velocity vectors, Devasenathipathy et al. (2003). It should be remarked that the use of this method presumes the presence of a steady flow, condition that is here ensured by the syringe pump. As far as the fully developed flow condition in the streamwise flow direction is concerned, this condition was verified by comparing the momentum flow rate imbalance at three distinctive streamwise locations (X = 200 m, X = 400 m and X = 700 m) against the value here considered to represent the accuracy of the micro-PIV velocity measurements. The momentum flow rate imbalance value was of the order of 3.4%, whereas the corresponding volumetric flow rate imbalance was 1.5%. Therefore, since the momentum flow rate imbalance value is affected by the volumetric flow rate one, its intrinsic value is considered to be less than 3%, i.e. within the experimental uncertainty. As a result, the microfluidic flow in the channel section under study is fully developed within a ca. 3% uncertainty.
6. Microchannel cross-section characterization using micro-PIV As pointed out in Section 4.2, one can ensure that the velocity field is measured with an accuracy of the order of the sub-pixel size under certain conditions. If the advantage of this precise velocity information could be used to characterize some particular spatial features, as for instance the solid boundaries, accuracies much higher than those provided by the direct visual measurements would be achieved. Taking into account that through an indirect approach, the kinematic boundary condition at a solid wall requires that the velocity field at the fluid/solid interface should not exhibit discontinuities, provided that the fluid can be modelled as a continuum (Gad-el-Hak, 1999) and the wall material is hydrophilic (Tretheway and Meinhart, 2002), the knowledge of the velocity field allows one to identify the presence of a wall as the region where the velocity is nil. However, because in the walls vicinity, micro-PIV measurements have limited accuracy, it is extremely difficult to directly measure the regions of nil velocity. In fact, very close to a solid interface, particles tend not to follow exactly the fluid motion since physical and chemical interactions between them and the walls start to take place. Moreover, due to the proximity of a medium with a different refractive index, i.e. the wall region corresponds to a liquid/solid interface, the process of imaging the particles motion close to the wall is also affected by optical phenomena, which are perceived mainly as severe optical distortions. To overcome these problems, it is preferable to use data from regions where the accuracy of the velocity measurements, and so the sub-pixel accuracy, can be guaranteed. Consequently, the velocity data measured sufficiently apart from the wall is extrapolated to regions near the wall holding the advantage of the sub-pixel accuracy. To achieve this objective, the velocity field has to be assumed to be defined by some analytical function V = V(x,y,z) which can be known from the yielded experimental data.
Since, as it was previously reported, the velocity field is fully developed, the field V = V(x,y,z) is assumed to exhibit a near parabolic variation along the cross-streamwise directions y and z and no variation in the streamwise flow direction x, which is typical of fully developed laminar channel flows. The Hagen–Poiseuille solution is only exact for flows inside circular tubes and between infinite parallel plates. For other simple non-circular cross-sectional shapes analytical solutions in terms of series expansions are available for the velocity profiles (Shah and London, 1978). For more complex cross-sectional shapes, as the one investigated in this work, there is no analytical solution to define the velocity profile. In spite of that, and although the mentioned complex series functions could be used, the use of simpler parabolic functions generally fits the experimental data in a much more accurate way, as observed by Stone et al. (2002). Summing up, for irregular geometries, as in the present case, where no analytical solutions are available, the most feasible choice, as experimental evidence of Fig. 5 suggests, is to use parabolic profiles to characterize V = V(x,y,z). Hence, the experimental velocity vectors at constant cross-streamwise flow direction are fitted to continuous parabolic functions of the type of U = Ay2 + By + C. The success of this approximation is revealed by the correlation coefficient of the parabolic fitting operation, which is always above 90%. The roots of this analytical function are computed and their values are associated with the location of the inner wall region. It should be reminded that this procedure assumes a no-slip velocity condition at the wall. However, in the case of a slip velocity this procedure remains valid provided that one knows the slip length. Since, in the present work, the static equilibrium contact angle is below 90◦ (ca. 74◦ ), i.e. the wall inner surface is hydrophilic, the no-slip velocity condition at the wall is valid. The characterization of the microchannel cross-section is carried out by repeating the procedure discussed above for the 61 measuring planes along the microchannel height. The results of the microchannel dimensions and crosssection are depicted in Fig. 7a and b. As it is observed, qualitatively, both methods yield a similar microchannel cross-section with dimensions of the same order of magnitude, the minimum/maximum width dimensions of ca. 142 m and ca. 843 m, respectively, and the height of approximately 900 m, yielding a microchannel hydraulic diameter of Dh = 637 m. In these measurement planes statistical variations in the measured velocities lead to a variation in the roots of the fitted parabolas. Therefore, in order to quantify the statistical variations in the velocity results supplied by micro-PIV, 3–5 realizations of the same measurement at the same horizontal plane were carried out and analysed. Since the flow is steady and laminar, temporal variations should not exist. Hence, the presence of any temporal statistical variations is interpreted as irregularities in the measurements. The horizontal error bars depicted in both Fig. 7a and b quantify this effect on the wall location value, which is predicted by the roots of the fitted parabolas. The horizontal error bars magnitude reveal the validity of the assumptions previously stated regarding the higher accuracy provided by micro-PIV. In average this accuracy is approximately 10 times higher. This result is not surprising if one observes a traditional microchannel image, see Fig. 4. As it is observed the inner wall region is very difficult to identify because of the image blurriness in this region. This feature leads to uncertainties much higher than the pixel size.
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
305
Fig. 7 – Microchannel cross-section and corresponding errors displayed as horizontal bars yielded by: (a) optical microscope; (b) micro-PIV measurements. Moreover, the intrinsic subjectivity of a visual inspection also contributes for the error depicted in Fig. 7a). As it would be expected, considering optical arguments, the occurrence of those facts increases with the top/bottom wall proximity. The micro-PIV results are also affected by this top/bottom wall proximity deterioration. In these measurement planes statistical variations in the measured velocities lead to a variation in the roots of the fitted parabolas. As a result, the accuracy in determining the wall location is decreased. This fact is revealed by the higher magnitude of the horizontal error bars in these regions. Despite of that, the effect of this error is not as severe as in the optical microscope case. The vertical error bars are not shown since their values are constant along the microchannel height and equal to ±11.5 m for the case of the optical microscope measurements, Fig. 7a), and ±17.1 m for the micro-PIV case, Fig. 7b). This figure reveals that, even though the micro-PIV yields better accuracy in the in-plane direction, the characterization of the microchannel cross-section along the out-of-plane direction is less precise with micro-PIV. However, the advantage of the micro-PIV procedure discussed herein in terms of the inplane plane accuracy pays off the slight disadvantage present in the out-of-plane direction.
7.
Conclusions
The relevance that geometrical parameters have in microscale flows associated with the relative inaccuracy/high cost of the experimental techniques currently available were the main motivations to develop this study. The method discussed is based on the use of the velocity data provided by the microPIV technique to establish the microchannel dimensions and its cross-sectional shape. To do so, 61 velocity profiles were measured along the microchannel height, which, in combination with the no-slip velocity condition at the wall, allowed for the reconstruction of the microchannel cross-section. The validity of the micro-PIV kinematic data was confirmed by comparing the volumetric flow rate supplied by the used syringe pump against that yielded by the integration of microPIV velocity profiles. The difference of 3.0% revealed the good quality of the micro-PIV results. The superiority of the micro-PIV technique in the accurate reconstruction of the microchannel boundaries was con-
firmed by comparing its results against those provided by raw visualizations using the standard optical microscope that is part of the experimental apparatus. With this procedure, it was verified that the use of the micro-PIV in combination with the methodology here applied yielded results, in average, 10 times more accurate. This result demonstrates that even when considerably large in-plane spatial resolutions are employed with micro-PIV, 23.68 m × 23.68 m as in the present study, the use of the velocity data can still be applied with success in the reconstruction of a microchannel cross-section allowing for the simultaneous study of the flow field and its boundaries.
References Adrian, R. and Yao, C., 1985, Pulsed laser technique application to liquid and gaseous flows and the scattering power of seed materials. Appl Opt, 24: 44–52. Bayraktar, T. and Pidugu, S., 2006, Characterization of liquid flows in microfluidic systems. Int J Heat Mass Transf, 49: 815–824. Celata, G., Cumo, M., McPhail, S. and Zummo, G., 2006, Characterization of fluid dynamic behaviour and channel wall effects in microtube. Int J Heat Fluid Flow, 27: 135–143. Delnoij, E., Westerweel, J., Deen, N., Kuipers, J. and van Swaaij, W., 1999, Ensemble correlation PIV applied to bubble plumes rising in a bubble column, In Proceedings of the 4th International Conference Gas–Liquid and Gas–Liquid–Solid Reactor Engineering (Delft, NL) Devasenathipathy, S., Santiago, J., Wereley, S., Meinhart, C. and Takehara, K., 2003, Particle imaging techniques for microfabricated fluidic systems. Exp Fluids, 34: 504–515. Gad-el-Hak, M., 1999, The fluid mechanics of microdevices—the Freeman Scholar lecture. J Fluids Eng, 121: 5–33. Guo, Z. and Li, Z., 2002, Size effect on micro-scale single phase phase flow and heat transfer, In Proceedings of the 12th International Heat Transfer Conference Hetsroni, G., Mosyak, A., Pogrebnyak, E. and Yarin, L.P., 2005, Fluid flow in micro-channels. Int J Heat Mass Transf, 48: 1982–1998. Inoué, S. and Spring, K., (1997). Video Microscopy (Second edition). (Plenum Press). Kandlikar, S., Garimella, S., Li, D., Colin, S. and King, M., (2006). Heat Transfer and Fluid Flow in Minichannels and MicroChannels (First edition). (Elsevier).
306
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
Keane, R. and Adrian, R., 1990, Optimization of particle image velocimeters. Part I. Double pulsed systems. Meas Sci Technol, 1: 1202–1215. Kinoshita, H., Oshima, M., Kaneda, S. and Fujii, T., 2005, Confocal micro-PIV measurements of internal flow in a moving droplet, In Proceeding of the 9th ICMSCLS , pp. 629–631. Lima, R., Wada, S., Tsubota, K. and Yamaguchi, T., 2006, Confocal micro-PIV measurements of three-dimensional profiles of cell suspension flow in a square microchannel. Meas Sci Technol, 17: 797–808. Magueijo, V., Semiao, V. and de Pinho, M.N., 2006, Effects of ultrafiltration permeation rates on the hydrodynamics of a minichannel/slit laminar flow. Chem Eng Sci, 61: 7139–7150. Meinhart, C., Wereley, S. and Santiago, J., 1999, PIV measurements of a microchannel flow. Exp Fluids, 27: 414–419. Meinhart, C., Wereley, S. and Santiago, J., 2000, A PIV algorithm for estimating time-average velocity fields. J Fluids Eng, 122: 285–297. Olsen, M. and Adrian, R., 2000, Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry. Exp Fluids, 29: 166–174. Prasad, A., Adrian, R., Landreth, C. and Offutt, P., 1992, Effect of resolution on the speed and accuracy of particle image velocimetry interrogation. Exp Fluids, 13: 105–116. Raffel, M., Willert, C. and Kompenhans, J., (1998). Particle Image Velocimetry—A Practical Guide. (Spinger-Verlag).
Santiago, J., Wereley, S., Meinhart, C., Beebe, D. and Adrian, R., 1998, A particle image velocimetry system for microfluidics. Exp Fluids, 25: 316–319. Shah, R.K. and London, A.L., (1978). Laminar Flow Forced Convection in Ducts, Volume Supp. 1 of Advances in Heat Transfer. (Academic Press). Silva, G., Leal, N. and Semiao, V., 2008, Micro-PIV and CFD characterization of flows in a microchannel: velocity profiles, surface roughness and Poiseuille numbers. Int J Heat Fluid Flow, 29: 1211–1220. Stone, S., Meinhart, C. and Wereley, S., 2002, A microfluidic-based nanoscope. Exp Fluids, 33: 613–619. Tretheway, D. and Meinhart, C., 2002, Fluid slip near hydrophobic microchannel walls. Phys Fluids, 14(3): L9–L12. Wereley, S.T. and Meinhart, C.D., (2004). Micron Resolution Particle Image Velocimetry, Diagnostic Techniques in microfluidics. (Springer-Verlag). Wereley, S., Gui, L. and Meinhart, C., 2002, Advanced algorithms for microscale particle image velocimetry. AIAA J, 40: 1047–1055. Westerweel, J., 1998, Fundamentals of digital particle image velocimetry. Meas Sci Technol, 8: 1379–1392. Westerweel, J., 2008, On velocity gradients in PIV interrogation. Exp Fluids, 44: 831–842. Willert, C. and Gharib, M., 1991, Digital particle image velocimetry. Exp Fluids, 10: 181–193.
Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
Determination of microchannels geometric parameters using micro-PIV Gonc¸alo Silva, Nuno Leal, Viriato Semiao ∗ Mechanical Engineering Department, Instituto Superior Tecnico, Universidade Tecnica de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
a b s t r a c t The precise characterization of microgeometries is a crucial task in the study of flow phenomena at these scales. Since the use of conventional optical microscopes is somehow limited in terms of accuracy, the present work studies the use of micro-PIV measurements to characterize a rather irregular microchannel cross-section. The micro-PIV is employed with a spatial resolution of 23.68 m × 23.68 m allowing for the location of the microchannel walls with an accuracy, in average, 10 times higher than that provided by the use of a conventional optical microscope. The accuracy of the micro-PIV results was validated by comparing the volumetric flow rate yielded by the integration of the micro-PIV velocity profiles against that supplied by the syringe pump employed in this experimental work. The 3% difference revealed the good quality of the measurements, demonstrating the potential of the micro-PIV technique to characterize both the flow kinematic parameters and the enclosing geometry. © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. Keywords: Micro-PIV; Microchannels; Geometry definition; Microfluidics
1.
Introduction
The study of fluid dynamics at micrometric scale geometries still presents conflicting and controversial results. Despite being nowadays established that decreasing the length scale makes relevant the phenomena traditionally neglected in macroscale flow theory, it is proved that the incorrect estimation of the experimental uncertainties can lead to misleading reports of those relevant phenomena, Gad-el-Hak (1999), Hetsroni et al. (2005), Bayraktar and Pidugu (2006) and Kandlikar et al. (2006). A simple error propagation analysis shows that, no matter what parameter one intends to measure, the overall experimental error increases due to the greater influence of the uncertainty in the geometrical characterization of the fluid flow domain as the length scale decreases (see, e.g. Hetsroni et al., 2005). Moreover, due to uncertainties in the manufacturing process and, sometimes, due to the lack of information from the manufacturer, the geometry dimensions and shape are generally not known to the desired accuracy.
The influence of an inaccurate geometrical characterization in microfluidic flow studies is not a new issue. In fact, its influence has repeatedly been reported as one of the main causes for obtaining incorrect results in experimental microfluidic studies, Hetsroni et al. (2005) and Magueijo et al. (2006). Pressure drop measurements in fully developed flows and its comparison against the results predicted by the conventional Fluid Mechanics theory applied to macroscale flows is perhaps the most notorious example of a situation where the flow geometry has a major influence in the accuracy of the final results. Distinctive examples of this situation can be found in the open literature. For instance, Guo and Li (2002) used a 40× optical microscope to characterize the inner diameter of a circular glass microtube obtaining a value of 84.7 m. The inclusion of this value in the pressure drop computation yielded values considerably higher than those predicted by the conventional Hagen–Poiseuille theory. The same measurements conducted with both a 400× optical microscope and a scanning electron microscope (SEM)
Abbreviations: CCD, charge coupled device; NA, numerical aperture; PIV, particle image velocimetry; SEM, scanning electron microscope; SPM, scanning probe microscope. ∗ Corresponding author. Tel.: +351 218417726; fax: +351 218475545. E-mail address: [email protected] (V. Semiao). Received 22 February 2008; Received in revised form 27 August 2008; Accepted 27 August 2008 0263-8762/$ – see front matter © 2008 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.cherd.2008.08.009
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
Nomenclature ddiff dp Dh e M n U V X Y zcorr Z
diameter of point spread function (m) particle diameter (m) microchannel hydraulic diameter (m) CCD pixel size (m) magnification refraction index fitted velocity profile (m/s) velocity vector (m/s) streamwise flow direction (m) cross-streamwise direction (width) (m) depth of correlation, half the thickness of the micro-PIV measurement volume (m) cross-streamwise direction (height) (m)
Greek symbols ız depth-of-field of a microscope objective (m) ε relative contribution of a particle displaced a distance z from the object plane wavelength of the emitted light (m)
revealed, for the same microtube, an inner diameter of 80 m, value that resulted in pressure drop values in good agreement with the theory. Another problem related to uncertainties in the geometrical characterization of the flow boundaries is described by Celata et al. (2006). These authors observed for a Teflon circular microtube with 304 m diameter the presence of slight discrepancies between the measured pressure drop and the value theoretically expected. The observation of SEM images of the microtube revealed deformations in the assumed cross-sectional circularity shape, i.e. instead of circular the microtube cross-section was best described by an ellipse. The comparison of the measured pressure drop data with the predicted Hagen–Poiseuille results, now for an elliptical cross-section, showed good matching. Based on the previous reasoning it is clear that optical microscopes and their conventional magnifications, usually below 100×, may not be particularly recommended if one needs to attain high accuracies. A related problem occurs when performing micro-PIV measurements. Since the micro-PIV technique is traditionally based on the use of optical microscopes, the visual determination of flow features is somehow limited by the accuracy of the optical system itself. To overcome this, avoiding the use of expensive equipment like the SEM or the scanning probe microscope (SPM), and achieving accuracies higher than those provided by a simple optical microscope, the micro-PIV velocity data can be used. This research work intends to perform a thorough study on the use of the micro-PIV technique to accurately characterize the microfluidic flow geometry, particularly its inner boundaries. The comparison of the results provided by micro-PIV measurements against those obtained through raw visual inspections with a conventional optical microscope is also performed in order to establish the validity of the former.
2.
Micro-PIV fundamentals
PIV is a non-intrusive optical technique used to measure the fluid motion. The motion of the fluid is determined by measuring the displacement of seeding particles in two recorded
299
images separated by a known period of time. Each image is divided into a certain number of cells, named interrogation areas, each of these containing a minimum amount of particles. For traditional cross-correlation algorithms, a minimum of five particles per interrogation area is usually required, whereas for more advanced correlation algorithms (e.g. ensemble time average algorithms) this requirement may not be necessary (Meinhart et al., 2000). The mean displacement of the amount of particles contained in each interrogation area is determined through a statistical correlation between the two images recorded. Micro-PIV is a technique that is based on the PIV concept. Yet, micro-PIV is applied to perform measurements in geometries with much smaller dimensions where much higher resolutions are required. Wereley et al. (2002) outlined three main aspects that differentiate micro-PIV from the traditional PIV: the flow field is volume illuminated instead of being illuminated by a laser light sheet, producing higher levels of background noise from out-of-focus particles if their density is too high; if particles are smaller than the laser light wavelength, which is needed in some studies where spatial resolutions of a few microns are required, light scattering techniques cannot be employed; Brownian motion of particles with diameters smaller than 1 m may compromise the evaluation of the flow velocity since that introduces an additional percentage of uncertainty in the final result. The first work where the PIV technique was successfully adapted for studying microscopic flows was carried out by Santiago et al. (1998). A pressure-driven Hele-Shaw with a cylindrical obstruction in the centre of a 120 m2 flow was studied and results were obtained with 3.45 m vectorto-vector distances with 6.9 m × 6.9 m spatial resolution. Later, Meinhart et al. (1999) measured the velocity field of a 30 m × 300 m glass channel with a spatial resolution of 0.9 m × 13.6 m in the wall normal and streamwise flow directions, respectively. Stone et al. (2002) used the micro-PIV technique applied to the study of flows in microchannels to determine the position of the microchannel walls within a spatial resolution of tens of nanometers for a single horizontal plane. The high accuracy of their measurements was obtained by using interrogation volumes of 0.9 m × 13.6 m × 1.8 m that already account for the existence of 50% overlap in the spanwise and streamwise directions. These authors were the first to propose the experimental technique discussed and developed in this work as an alternative to the application of specific microscopes that, as mentioned before, possess limitations for measurements in liquid filled microchannels. Such technique is applied herein at 61 horizontal measurement planes to reconstruct the entire microchannel cross-section.
3.
Experimental apparatus
In this work a microchannel manufactured by DantecDynamics® was used, which is shown in Fig. 1. This microchannel is manufactured by laser ablation in a polymeric chip (PMMA) and sealed with the same material. This ensemble allows for micro-PIV measurements with reduced optical aberrations. The study was performed in the rectilinear region of the microchannel, denoted in Fig. 1 by the zone enclosed by the elliptical line. The pressure required for the fluid to flow was provided by a syringe pump (NE-1000, New Era Pump Systems, Inc.® ). Error tests performed to this pump indicated the uncertainty
300
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
Table 1 – Maximum spatial resolution of commonly used air-immersion lenses
Fig. 1 – The studied microchannel integrated in a chip platform supplied by DantecDynamics® . of ±1.5 l/min. The fluid used in all experiments was deionised water. A micro-PIV system was used to study the fluid flow inside the above microchannel. A pulsed Nd:YAG laser New Wave® Solo II-15 ( = 532 nm) was used to illuminate the measurement section of the microchannel. Fluorescent nile red flow-tracing particles with 1 m of average diameter were added to deionised water in a proportion of 0.5 l (of a solution of 3.64 × 1010 particles/ml) to 1 ml of water. An epi-fluorescent filter was used to prevent the background noise having wavelength below the red light spectrum ( = 575 nm) to reach the Flowsense® 2M CCD (charge coupled device) camera. This camera has a 1600 × 1186 pixel2 resolution (pixel size = 7.4 m) with 8/10-bit intensity resolution. Time delays ranging from 100 s to 500 s between frames and a repetition rate of 15 Hz were used. In order to obtain magnified particle images the CCD camera was coupled to a Leica® DM ILM microscope with an air immersion lens of magnification M = 10 and numerical aperture of NA = 0.25. A schematic representation of the micro-PIV system used in this work is presented in Fig. 2.
4.
Measurement resolutions
4.1.
Optical microscope resolution
The images of the microchannel geometry are captured by a CCD camera connected to an optical microscope. Therefore, in a micro-PIV setup, the image properties are influenced by both the microscope and the CCD camera optical parameters. The maximum in-plane spatial resolution of any optical system is limited by light diffraction, a phenomenon consist-
M
NA
ddiff (m)
Maximum spatial resolution (m)
5 10 20 40 63 100
0.12 0.25 0.40 0.55 0.70 0.90
54.1 51.9 64.9 94.4 116.8 144.2
27.05 25.95 32.45 47.20 58.40 72.10
ing of an interference pattern that images the object as a central disk (the Airy disk) surrounded by concentric rings (the Airy rings), as for reported for instance by Inoué and Spring (1997) and Raffel et al. (1998). It can be demonstrated, using the so-called Fraunhofer approximation (Adrian and Yao, 1985), that the diameter of the Airy disk in the image plane, ddiff , is expressed by Eq. (1), where M is the lens magnification, is the wavelength of the emitted light and NA the lens numerical aperture: ddiff = 2.44 M
2 NA
(1)
The resolution of an optical system can thus be defined as 0.5ddiff , i.e. half the distance between two adjacent Airy disk centres. The definition of the spatial resolution by this criterion is known as the Rayleigh criterion (Inoué and Spring, 1997). Taking into account that the image in a CCD camera is formed by an array of pixels, the pixel size emerges as another limiting parameter to the maximum optical resolution since no geometry can be resolved below the pixel dimension. However, considering the pixel size of a CCD camera, which is usually of the order of 10 m or less, and the optical characteristics of commonly used air-immersion objectives, the maximum in-plane spatial resolution in a micro-PIV is limited by the light diffraction phenomenon. Table 1 illustrates several diffraction and resolution values for commonly used air-immersion lenses. Besides the maximum in-plane spatial resolution one also needs to take into consideration the out-of-plane resolution limit. The out-of-plane resolution is given by the depth-of-field of the optical system, ıZ , which can be viewed as an estimate of the distance that a microscope slide may be moved in the
Fig. 2 – Scheme of the DantecDynamics® assembled micro-PIV equipment.
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
4.2.
Table 2 – Depth-of-field of commonly used air-immersion lenses. M
NA
ız (m)
5 10 20 40 63 100
0.12 0.25 0.40 0.55 0.70 0.90
49.3 11.5 4.3 2.1 1.3 0.7
vertical direction while still maintaining focus on an infinitely thin specimen. According to Inoué and Spring (1997), the depth-of-field of a microscope objective is given by the sum of the focal depth due to diffraction, the first term on right hand side of Eq. (2), and the focal depth due to geometrical effects, the second term on right hand side of Eq. (2): ız =
n0 NA2
+
ne NA M
(2)
In the previous equation, n is the refractive index of the imaging medium, 0 is the wavelength of the imaged light in vacuum and e is the smallest distance that can be resolved by the image detector placed in the microscope’s image plane, which corresponds to the CCD pixel size in the present case. Table 2 illustrates the out-of-plane resolution achieved by commonly used air-immersion lenses with n = 1.00, 0 = 532 nm and e = 7.4 m.
301
Micro-PIV resolution
As previously stated, in the micro-PIV technique the image is divided into a certain number of cells named interrogation areas. To make full use of the information provided by the images, the interrogation areas are generally overlapped. As a result, and although each velocity vector represents the average velocity of the flow inside the interrogation area, the distance between neighbouring velocity vectors is greatly reduced due to the overlapping process. The distance between neighbouring velocity vectors is considered here to define the in-plane spatial resolution of the micro-PIV measurements. The velocity vector yielded by the micro-PIV technique characterizes the average motion of the flow or, more precisely, the average motion of the total amount of particles in that region within the interrogation area. Excluding the presence of severe velocity gradients, which may significantly deteriorate the correlation peak biasing the velocity measurement (Westerweel, 2008), as long as all particles stay within the interrogation area, the computed velocity vector is always an accurate measure of the mean flow velocity within the interrogation area. However, by construction, the micro-PIV technique always places the measured velocity vector in the centre of the interrogation area. This can lead to significant bias errors in the velocity representation, even in the case of minor velocity gradients. This error exists since one is assuming that the mean velocity is also located at the interrogation area centre. However, this assumption is only true when there is no velocity gradient or, when the velocity gradient exists but
Fig. 3 – Velocity profile within an interrogation area and corresponding histogram of velocities: (a) linear velocity profile, (b) histogram of velocities for a linear velocity profile, (c) parabolic velocity profile and (d) histogram of velocities for a parabolic velocity profile.
302
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
without compromising the overall results quality in terms of valid/rejected vectors, this spatial resolution was intentionally chosen to demonstrate that, provided the conditions previously discussed are verified, the method studied herein is applicable. Regarding the spatial resolution in the out-of-plane direction, since the flow field in micro-PIV is volume illuminated, this parameter is not just simply given by the lens depthof-field. This is so because there are particles that still emit enough light to contribute to the correlation signal despite being out-of-focus. Therefore, the resolution in the vertical direction, named by Olsen and Adrian (2000) as depth-ofcorrelation, is defined as twice the distance that a particle of diameter dp can be positioned from the object plane so that the intensity along the optical axis is an arbitrarily specified fraction of its focused intensity, ε. Beyond this distance, the particle’s intensity is sufficiently low and does not influence the velocity measurement. For this reason, the depth-ofcorrelation, 2Zcorr , can be viewed as the limiting out-of-plane spatial resolution of a micro-PIV measurement. According to the study of Olsen and Adrian (2000) the depth-of-correlation is given by the following equation:
it is constant, i.e. when the velocity changes linearly within the interrogation area, see Fig. 3a and the corresponding histogram, Fig. 3b. For non-linear velocity variations, like those sketched in Fig. 3c, the bias velocity error is present and is represented in Fig. 3d. Consequently, if one assumes that the presence of a velocity gradient does not lead to out-of-pair particles, the error produced by such gradient is only relevant when the interrogation area is so large that the inaccuracy resulting from the linear velocity assumption within the area is significant. Consequently, if the real velocity profile can be represented, without significant loss of accuracy, by piecewise linear functions (where the length of each section corresponds to the interrogation area length) the assumption of negligible velocity gradients is valid. This analysis leads to the conclusion that small interrogation areas provide the most accurate results since they minimize this error. However, considering the fact that the signal strength in the velocity computation process is proportional to the number of particle pairs within the interrogation area (Keane and Adrian, 1990; Westerweel, 1998), if one keeps the particles density constant it immediately follows that the interrogation areas size cannot be decreased below
2zcorr = 2
√ 1− ε √ ε
a certain dimension. For that reason, the interrogation area size must be optimally chosen so that the maximal signal strength is achieved with the less velocity gradient error contribution possible. A rule-of-thumb generally followed by the PIV community consists in choosing interrogation areas sizes that simultaneously ensures that, at least, five particle pairs are present (Keane and Adrian, 1990) and that the particles displacement does not exceed 1/4 of the interrogation area length (Keane and Adrian, 1990; Westerweel, 1998; Raffel et al., 1998). If the velocity gradient contribution can be neglected or, at least, if the velocity vector location within the interrogation area is not too inaccurate, the use of sub-pixel interpolation schemes (Willert and Gharib, 1991; Prasad et al., 1992) allows one to estimate the average velocity value inside the interrogation area to sub-pixel accuracy. In fact, as demonstrated by Prasad et al. (1992), as long as particles are resolved by 2–3 pixels, the accuracy in locating the maximum peak in the correlation plane, i.e. the precision in determining the velocity vector within the interrogation area is approximately 0.1 pixels. Therefore, even though the micro-PIV in-plane spatial resolution is usually of the order of several pixels, the kinematic information supplied by this technique, which is given by the mean velocity vector inside the interrogation area, is of the order of 0.1 pixels. If the information provided by the kinematic field can be somehow related to the determination of some spatial feature in the flow field, one will be able to spatially characterize that feature with accuracy higher than that provided by a conventional optical microscopy technique. Therefore, in order to achieve an in-plane spatial resolution of less than one pixel size, while ensuring that this resolution provides an accurate measure of the flow field, the presence of velocity gradients within the interrogation areas cannot be significant. In this study, and because the flow is fully developed, as described below, the in-plane spatial resolution was chosen to have a value of 23.68 m × 23.68 m, with 50% overlap, in a field of view of 1172 m × 887 m. Although the in-plane spatial resolution could be significantly decreased
2
dp 2 (n/NA) − 1 4
2
+
2
2 1/2
5.95(M + 1) 2 (n/NA) − 1 4M2
(3)
Since the depth-of-correlation value is always larger than the depth-of-field value of the used lens, the limiting outof-plane resolution of a micro-PIV measurement is always larger than that of a conventional microscope measurement. Table 3 illustrates common values of depth-of-correlations with air-immersion lenses assuming dp = 1 m and ε = 0.01 and absolute differences to corresponding depth-of-field values, z = zcorr − ız . As it can be seen in Table 3 the micro-PIV spatial resolutions in the out-of-plane direction are considerably larger than those provided by conventional microscopy measurements. This difference decreases significantly as the lens magnification and numerical aperture increase. To overcome this problem the scientific community has associated the micro-PIV technique with more advanced microscopy techniques. The use of laser scanning confocal microscopy (LSCM) is one of the most recognized examples. The use of such microscopes allows one to achieve high out-of-plane resolutions without compromising the in-plane spatial resolution, i.e. without having to use high magnifications. For instance, Kinoshita et al. (2005), using a micro-PIV system with a confocal microscope, performed velocity measurements in moving droplets inside microchannels with a field of view of 228 m × 171 m and a confocal depth of 1.8 m. The major drawbacks of using the confocal technique
Table 3 – Depth-of-correlation of commonly used air-immersion lenses assuming dp = 1 m and ε = 0.01 and absolute differences to corresponding depth-of-field values M
NA
2Zcorr (m)
5 10 20 40 63 100
0.12 0.25 0.40 0.55 0.70 0.90
161.9 34.2 12.8 6.5 3.7 1.5
z (m) ±31.65 ±5.60 ±2.10 ±1.15 ±0.55 ±0.05
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
are the considerably higher cost of the equipment required and the relatively lower state-of-development, when compared to the conventional micro-PIV technique. This fact is denoted by the limited range of applications that can use confocal micro-PIV with success at present (Lima et al., 2006). Despite of these drawbacks, the virtues of confocal micro-PIV are undeniable, particularly in situations where measurements with high out-of-plane resolutions are crucial. Similarly to the analysis performed for the in-plane spatial resolution, the out-of-plane resolution size, i.e. the thickness of the measurements control volume, can be considered adequate as long as the displacement of the particles in the out-of-plane direction does not exceed 1/4 of the control volume thickness and also that, within the interrogation area, severe velocity gradients in this out-of-plane direction do not occur. The lens used in this work has M = 10 and NA = 0.25, which corresponds to a depth-of-correlation value of 34.2 m (see Table 3). Therefore, in order to perform the reconstruction of the three dimensional velocity field with a vertical overlap approximately equal to the horizontal one, i.e. 50%, the horizontal planes were measured each 15 m in the vertical direction. This vertical motion is controlled by a stepper motor with reproducibility better than 1 m and a total precision of ±3 m that is controlled by a real time RISC processor. The stepper motor can be addressed with a resolution of 1.5 nm. The influence of the chosen in-plane and out-of-plane resolution values in the accurate description of the velocity field is performed below.
303
Fig. 5 – Measured velocity profile at X = 200 m of a horizontal measurement plane at Z = 500 m.
5. Estimation of the micro-PIV velocity uncertainty and validity of the flow fully developed condition The a priori estimation of the error associated with the velocity measurement with micro-PIV is a complex task since each micro-PIV realization comprises the participation of several distinctive parameters whose interaction, in practice, is almost impossible to quantify in detail (a deeper analysis of some of the most common PIV error sources can be found, for instance, in Raffel et al., 1998 or Wereley and Meinhart, 2004). Consequently, and in order to have an order of magnitude of the micro-PIV error present in the overall velocity measurements, an approach similar to one described in Silva et al. (2008) is adopted here. With this approach several measurement planes of the velocity field, as the one depicted in Fig. 4,
Fig. 4 – Example of an image of the studied microchannel at a horizontal plane Z = 200 m with measured velocity vectors.
Fig. 6 – Three-dimensional velocity profile built up from 61 two-dimensional horizontal velocity profiles, similar to that depicted in Fig. 5. are extracted along the microchannel height. In each plane Z = const, and for a constant streamwise location, X = 200 m in the present case, the velocity profile is computed. Fig. 5 shows an example of such velocity profile at Z = 500 m and X = 200 m. This procedure, applied to all the 61 horizontal measurement planes, allows for the reconstruction of the overall three-dimensional velocity profile as depicted in Fig. 6. The volumetric flow rate given by the volume enclosed by this three-dimensional velocity profile, which is computed through numerical integration, is compared against the corresponding flow rate supplied by the syringe pump, which is considered to provide a theoretical value. The difference between flow rates provides an indication of the bias error of the micro-PIV technique in the case under study. For the volumetric flow rate used herein, which is ca. 1 ml/min, the difference between the micro-PIV predicted value and the one supplied by the syringe pump, is ca. 3%. This indicates that, in average, the velocity information provided by the microPIV technique is expected to have an accuracy of ca. 3%. Since the micro-PIV data used in this analysis was obtained with the in-plane and out-of-plane resolutions described in Section 4.2, the accuracy of these resolutions is also denoted by
304
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
ca. 3% error. It should be stressed that the computation of the velocity field was performed with an ensemble time average correlation algorithm, as described by Meinhart et al. (2000) and Delnoij et al. (1999), in order to obtain a velocity field with the less unphysical velocity vectors. These unphysical vectors can be caused by the presence of both a low signalto-noise ratio in the correlation plane, caused by the required low seeding concentration, and the influence of the Brownian motion on the computation of the instantaneous velocity vectors, Devasenathipathy et al. (2003). It should be remarked that the use of this method presumes the presence of a steady flow, condition that is here ensured by the syringe pump. As far as the fully developed flow condition in the streamwise flow direction is concerned, this condition was verified by comparing the momentum flow rate imbalance at three distinctive streamwise locations (X = 200 m, X = 400 m and X = 700 m) against the value here considered to represent the accuracy of the micro-PIV velocity measurements. The momentum flow rate imbalance value was of the order of 3.4%, whereas the corresponding volumetric flow rate imbalance was 1.5%. Therefore, since the momentum flow rate imbalance value is affected by the volumetric flow rate one, its intrinsic value is considered to be less than 3%, i.e. within the experimental uncertainty. As a result, the microfluidic flow in the channel section under study is fully developed within a ca. 3% uncertainty.
6. Microchannel cross-section characterization using micro-PIV As pointed out in Section 4.2, one can ensure that the velocity field is measured with an accuracy of the order of the sub-pixel size under certain conditions. If the advantage of this precise velocity information could be used to characterize some particular spatial features, as for instance the solid boundaries, accuracies much higher than those provided by the direct visual measurements would be achieved. Taking into account that through an indirect approach, the kinematic boundary condition at a solid wall requires that the velocity field at the fluid/solid interface should not exhibit discontinuities, provided that the fluid can be modelled as a continuum (Gad-el-Hak, 1999) and the wall material is hydrophilic (Tretheway and Meinhart, 2002), the knowledge of the velocity field allows one to identify the presence of a wall as the region where the velocity is nil. However, because in the walls vicinity, micro-PIV measurements have limited accuracy, it is extremely difficult to directly measure the regions of nil velocity. In fact, very close to a solid interface, particles tend not to follow exactly the fluid motion since physical and chemical interactions between them and the walls start to take place. Moreover, due to the proximity of a medium with a different refractive index, i.e. the wall region corresponds to a liquid/solid interface, the process of imaging the particles motion close to the wall is also affected by optical phenomena, which are perceived mainly as severe optical distortions. To overcome these problems, it is preferable to use data from regions where the accuracy of the velocity measurements, and so the sub-pixel accuracy, can be guaranteed. Consequently, the velocity data measured sufficiently apart from the wall is extrapolated to regions near the wall holding the advantage of the sub-pixel accuracy. To achieve this objective, the velocity field has to be assumed to be defined by some analytical function V = V(x,y,z) which can be known from the yielded experimental data.
Since, as it was previously reported, the velocity field is fully developed, the field V = V(x,y,z) is assumed to exhibit a near parabolic variation along the cross-streamwise directions y and z and no variation in the streamwise flow direction x, which is typical of fully developed laminar channel flows. The Hagen–Poiseuille solution is only exact for flows inside circular tubes and between infinite parallel plates. For other simple non-circular cross-sectional shapes analytical solutions in terms of series expansions are available for the velocity profiles (Shah and London, 1978). For more complex cross-sectional shapes, as the one investigated in this work, there is no analytical solution to define the velocity profile. In spite of that, and although the mentioned complex series functions could be used, the use of simpler parabolic functions generally fits the experimental data in a much more accurate way, as observed by Stone et al. (2002). Summing up, for irregular geometries, as in the present case, where no analytical solutions are available, the most feasible choice, as experimental evidence of Fig. 5 suggests, is to use parabolic profiles to characterize V = V(x,y,z). Hence, the experimental velocity vectors at constant cross-streamwise flow direction are fitted to continuous parabolic functions of the type of U = Ay2 + By + C. The success of this approximation is revealed by the correlation coefficient of the parabolic fitting operation, which is always above 90%. The roots of this analytical function are computed and their values are associated with the location of the inner wall region. It should be reminded that this procedure assumes a no-slip velocity condition at the wall. However, in the case of a slip velocity this procedure remains valid provided that one knows the slip length. Since, in the present work, the static equilibrium contact angle is below 90◦ (ca. 74◦ ), i.e. the wall inner surface is hydrophilic, the no-slip velocity condition at the wall is valid. The characterization of the microchannel cross-section is carried out by repeating the procedure discussed above for the 61 measuring planes along the microchannel height. The results of the microchannel dimensions and crosssection are depicted in Fig. 7a and b. As it is observed, qualitatively, both methods yield a similar microchannel cross-section with dimensions of the same order of magnitude, the minimum/maximum width dimensions of ca. 142 m and ca. 843 m, respectively, and the height of approximately 900 m, yielding a microchannel hydraulic diameter of Dh = 637 m. In these measurement planes statistical variations in the measured velocities lead to a variation in the roots of the fitted parabolas. Therefore, in order to quantify the statistical variations in the velocity results supplied by micro-PIV, 3–5 realizations of the same measurement at the same horizontal plane were carried out and analysed. Since the flow is steady and laminar, temporal variations should not exist. Hence, the presence of any temporal statistical variations is interpreted as irregularities in the measurements. The horizontal error bars depicted in both Fig. 7a and b quantify this effect on the wall location value, which is predicted by the roots of the fitted parabolas. The horizontal error bars magnitude reveal the validity of the assumptions previously stated regarding the higher accuracy provided by micro-PIV. In average this accuracy is approximately 10 times higher. This result is not surprising if one observes a traditional microchannel image, see Fig. 4. As it is observed the inner wall region is very difficult to identify because of the image blurriness in this region. This feature leads to uncertainties much higher than the pixel size.
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
305
Fig. 7 – Microchannel cross-section and corresponding errors displayed as horizontal bars yielded by: (a) optical microscope; (b) micro-PIV measurements. Moreover, the intrinsic subjectivity of a visual inspection also contributes for the error depicted in Fig. 7a). As it would be expected, considering optical arguments, the occurrence of those facts increases with the top/bottom wall proximity. The micro-PIV results are also affected by this top/bottom wall proximity deterioration. In these measurement planes statistical variations in the measured velocities lead to a variation in the roots of the fitted parabolas. As a result, the accuracy in determining the wall location is decreased. This fact is revealed by the higher magnitude of the horizontal error bars in these regions. Despite of that, the effect of this error is not as severe as in the optical microscope case. The vertical error bars are not shown since their values are constant along the microchannel height and equal to ±11.5 m for the case of the optical microscope measurements, Fig. 7a), and ±17.1 m for the micro-PIV case, Fig. 7b). This figure reveals that, even though the micro-PIV yields better accuracy in the in-plane direction, the characterization of the microchannel cross-section along the out-of-plane direction is less precise with micro-PIV. However, the advantage of the micro-PIV procedure discussed herein in terms of the inplane plane accuracy pays off the slight disadvantage present in the out-of-plane direction.
7.
Conclusions
The relevance that geometrical parameters have in microscale flows associated with the relative inaccuracy/high cost of the experimental techniques currently available were the main motivations to develop this study. The method discussed is based on the use of the velocity data provided by the microPIV technique to establish the microchannel dimensions and its cross-sectional shape. To do so, 61 velocity profiles were measured along the microchannel height, which, in combination with the no-slip velocity condition at the wall, allowed for the reconstruction of the microchannel cross-section. The validity of the micro-PIV kinematic data was confirmed by comparing the volumetric flow rate supplied by the used syringe pump against that yielded by the integration of microPIV velocity profiles. The difference of 3.0% revealed the good quality of the micro-PIV results. The superiority of the micro-PIV technique in the accurate reconstruction of the microchannel boundaries was con-
firmed by comparing its results against those provided by raw visualizations using the standard optical microscope that is part of the experimental apparatus. With this procedure, it was verified that the use of the micro-PIV in combination with the methodology here applied yielded results, in average, 10 times more accurate. This result demonstrates that even when considerably large in-plane spatial resolutions are employed with micro-PIV, 23.68 m × 23.68 m as in the present study, the use of the velocity data can still be applied with success in the reconstruction of a microchannel cross-section allowing for the simultaneous study of the flow field and its boundaries.
References Adrian, R. and Yao, C., 1985, Pulsed laser technique application to liquid and gaseous flows and the scattering power of seed materials. Appl Opt, 24: 44–52. Bayraktar, T. and Pidugu, S., 2006, Characterization of liquid flows in microfluidic systems. Int J Heat Mass Transf, 49: 815–824. Celata, G., Cumo, M., McPhail, S. and Zummo, G., 2006, Characterization of fluid dynamic behaviour and channel wall effects in microtube. Int J Heat Fluid Flow, 27: 135–143. Delnoij, E., Westerweel, J., Deen, N., Kuipers, J. and van Swaaij, W., 1999, Ensemble correlation PIV applied to bubble plumes rising in a bubble column, In Proceedings of the 4th International Conference Gas–Liquid and Gas–Liquid–Solid Reactor Engineering (Delft, NL) Devasenathipathy, S., Santiago, J., Wereley, S., Meinhart, C. and Takehara, K., 2003, Particle imaging techniques for microfabricated fluidic systems. Exp Fluids, 34: 504–515. Gad-el-Hak, M., 1999, The fluid mechanics of microdevices—the Freeman Scholar lecture. J Fluids Eng, 121: 5–33. Guo, Z. and Li, Z., 2002, Size effect on micro-scale single phase phase flow and heat transfer, In Proceedings of the 12th International Heat Transfer Conference Hetsroni, G., Mosyak, A., Pogrebnyak, E. and Yarin, L.P., 2005, Fluid flow in micro-channels. Int J Heat Mass Transf, 48: 1982–1998. Inoué, S. and Spring, K., (1997). Video Microscopy (Second edition). (Plenum Press). Kandlikar, S., Garimella, S., Li, D., Colin, S. and King, M., (2006). Heat Transfer and Fluid Flow in Minichannels and MicroChannels (First edition). (Elsevier).
306
chemical engineering research and design 8 7 ( 2 0 0 9 ) 298–306
Keane, R. and Adrian, R., 1990, Optimization of particle image velocimeters. Part I. Double pulsed systems. Meas Sci Technol, 1: 1202–1215. Kinoshita, H., Oshima, M., Kaneda, S. and Fujii, T., 2005, Confocal micro-PIV measurements of internal flow in a moving droplet, In Proceeding of the 9th ICMSCLS , pp. 629–631. Lima, R., Wada, S., Tsubota, K. and Yamaguchi, T., 2006, Confocal micro-PIV measurements of three-dimensional profiles of cell suspension flow in a square microchannel. Meas Sci Technol, 17: 797–808. Magueijo, V., Semiao, V. and de Pinho, M.N., 2006, Effects of ultrafiltration permeation rates on the hydrodynamics of a minichannel/slit laminar flow. Chem Eng Sci, 61: 7139–7150. Meinhart, C., Wereley, S. and Santiago, J., 1999, PIV measurements of a microchannel flow. Exp Fluids, 27: 414–419. Meinhart, C., Wereley, S. and Santiago, J., 2000, A PIV algorithm for estimating time-average velocity fields. J Fluids Eng, 122: 285–297. Olsen, M. and Adrian, R., 2000, Out-of-focus effects on particle image visibility and correlation in microscopic particle image velocimetry. Exp Fluids, 29: 166–174. Prasad, A., Adrian, R., Landreth, C. and Offutt, P., 1992, Effect of resolution on the speed and accuracy of particle image velocimetry interrogation. Exp Fluids, 13: 105–116. Raffel, M., Willert, C. and Kompenhans, J., (1998). Particle Image Velocimetry—A Practical Guide. (Spinger-Verlag).
Santiago, J., Wereley, S., Meinhart, C., Beebe, D. and Adrian, R., 1998, A particle image velocimetry system for microfluidics. Exp Fluids, 25: 316–319. Shah, R.K. and London, A.L., (1978). Laminar Flow Forced Convection in Ducts, Volume Supp. 1 of Advances in Heat Transfer. (Academic Press). Silva, G., Leal, N. and Semiao, V., 2008, Micro-PIV and CFD characterization of flows in a microchannel: velocity profiles, surface roughness and Poiseuille numbers. Int J Heat Fluid Flow, 29: 1211–1220. Stone, S., Meinhart, C. and Wereley, S., 2002, A microfluidic-based nanoscope. Exp Fluids, 33: 613–619. Tretheway, D. and Meinhart, C., 2002, Fluid slip near hydrophobic microchannel walls. Phys Fluids, 14(3): L9–L12. Wereley, S.T. and Meinhart, C.D., (2004). Micron Resolution Particle Image Velocimetry, Diagnostic Techniques in microfluidics. (Springer-Verlag). Wereley, S., Gui, L. and Meinhart, C., 2002, Advanced algorithms for microscale particle image velocimetry. AIAA J, 40: 1047–1055. Westerweel, J., 1998, Fundamentals of digital particle image velocimetry. Meas Sci Technol, 8: 1379–1392. Westerweel, J., 2008, On velocity gradients in PIV interrogation. Exp Fluids, 44: 831–842. Willert, C. and Gharib, M., 1991, Digital particle image velocimetry. Exp Fluids, 10: 181–193.