Ray tracing Particle Image Velocimetry – Challenges in the application to a packed bed

doi:10.21203/rs.3.rs-2213411/v1

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Ray tracing Particle Image Velocimetry – Challenges in the application to a packed bed

https://doi.org/10.21203/rs.3.rs-2213411/v1

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published 16 Jun, 2023

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Ray tracing Particle Image Velocimetry (RT-PIV) is a novel technique for high resolution velocity measurements in challenging optical systems, such as transparent packed beds, that uses ray tracing to correct for distortions introduced by transparent geometries. The ray tracing based correction is a post processing step applied to PIV particle images before classical PIV evaluation. In this study, RT-PIV is performed in the top layer of a body centred cubic (bcc) sphere packing with gaseous flow, where optical access is generated by the use of transparent N-BK7 glass balls with a diameter of d = 40mm. RT-PIV introduces new challenges, for example a limited field of view, illumination difficulties, a very large required depth of field and high sensitivity to geometric parameters used in the ray tracing correction. These challenges are discussed in the present work as well as the validation of the ray tracing reconstruction step and the finally obtained corrected vector fields.

Ray Tracing Particle Image Velocimetry (RT-PIV)

transparent spherical particles

packed bed

distortion correction

gas flow

Packed beds with gaseous flows are part of many industrial processes which, in their majority, come with an immense energy consumption. Packed beds are, for example, used in the food industry, for instance for coffee roasting, for drying processes in several fields, and for catalytic processes in chemical engineering. For yield and product quality optimization and the reduction of resource and energy consumption, a thorough understanding of the processes inside the packed bed, such as heat and mass transfer, is indispensable. The heat and mass transfer are thereby dependent on the characteristics of the gaseous flow in the bed. The measurement of flow fields in the complex geometry is, however, very challenging.

To measure gas velocities in packed beds, intrusive measurement techniques like hot-wire anemometry or endoscopic PIV can be applied. Different authors (Cairns and Prausnitz 1959; Morales et al. 1951; Moran and Glicksman 2003; Schwartz and Smith 1953) worked with hot-wire anemometry to obtain the velocity distribution for different bed heights. However, only pointwise measurements are obtained with this approach, so that numerous anemometers have to be applied at the same time to obtain velocity information at several points, or the measurement position has to be varied in a stationary flow. Both lead to an impact of the inserted measurement device on the gas flow. 2D-velocity fields in the interstices of packed beds can also be obtained by endoscopic PIV measurements (Blois et al. 2012; Shaffer et al. 2013). Unfortunately, the macroscopic endoscopes also modify the flow field.

Non-intrusive measurement techniques avoid influencing the flow. Tomographic techniques can be used, like magnetic resonance imaging (Baldwin et al. 1996; Kuniyasu Ogawa et al. 2001; Lovreglio et al. 2018; Ogawa et al. 2000; Ogawa et al. 2001; Ren et al. 2005; Sankey et al. 2009; Sederman et al. 1998; Suekane et al. 2003) or particle emission tomography (Khalili et al. 1998). These approaches do not require transparent geometries but suffer from low spatial and temporal resolutions and materials incompatibilities (Gladden et al. 2006; Poelma 2020).

Optical measurement techniques allow for high resolution in space and time. These, however, require optical access. Therefore, PIV measurements were primarily performed in near wall regions of packed beds (Wu and Mirbod 2018) or on the surface of beds (Pokrajac and Manes 2009) where one has direct optical access. By the use of transparent materials, e.g. glass balls with a refractive index in the range of n = 1.45 to n = 1.96 (Budwig 1994), optical access to the packed bed can also be generated. Unfortunately, transparent packing materials introduce distortions in the PIV images, since the material and the flow have different refractive indices. In general, the distortions can be avoided with refractive index matching (RIM) using different index matched fluids (Borrero-Echeverry and Morrison 2016; Budwig 1994; Wiederseiner et al. 2011). RIM has also been applied to Particle Tracking Velocimetry (PTV) and PIV techniques in porous media or packed beds (Huang et al. 2008; Monica et al. 2009; Moroni and Cushman 2001). In this way, turbulence quantities of the interstitial flow have been determined (Khayamyan et al. 2017a; Khayamyan et al. 2017b; Patil and Liburdy 2013, 2015) and first attempts to measure 3D velocity fields have been done via tomographic PIV (Larsson et al. 2018). This approach is, however, limited to liquids as working fluid because of the significant discrepancy in the refractive indices between gases and solids so that no matching is possible. For gaseous flows, similarity approaches have therefore been applied for data evaluation and interpretation (Hassan and Dominguez-Ontiveros 2008; Nguyen et al. 2019).

To the best of our knowledge, velocity data of gaseous flows in packed beds, measured by PIV, has not been obtained systematically up to now. First approaches have been done by our group (Kováts et al. 2015) and the fundamentals of the technique proposed in this paper have been published in (Martins et al. 2018a, b). Besides RT-PIV, all aforementioned measurement techniques have to accept compromises between influencing the flow, use of liquids instead of gases, and temporal and/or spatial resolution.

Therefore, RT-PIV is proposed here for the measurement of instantaneous 2D gas velocity fields inside transparent packed beds. RT-PIV is a non-intrusive optical measurement technique and allows for temporally as well as spatially highly resolved measurements of gaseous flows in packed beds.

The ray tracing correction takes as input a 3D computer model of the experimental setup as well as the raw PIV images. Based on it, two optical light transport simulations are performed. In the first one, the captured PIV image is used as source and an inverse simulation from the camera into the experimental setup recovers an approximation of the light field. The second step simulates the image formation in the experimental setup but with the transparent geometry removed, yielding an undistorted image.

Very few experimental realisations of RT-PIV exist up to now. They are mostly in the field of surface reconstruction or correcting distortions in PIV measurements caused by liquid-gas boundaries (Kang et al. 2004; Minor et al. 2007; Wu et al. 2019). Some applications of ray tracing in PIV measurements can also be found in the field of in-cylinder flows (Daher et al. 2019; Zha et al. 2016) where curvature effects due to transparent cylindrical walls require correction.

In the present work, we discuss the application of RT-PIV in gaseous flows through a packed bed, with particular emphasis on the challenges and difficulties the approach implies. Furthermore, the step-wise validation of the ray tracing based correction method is presented. As setup we consider 2D RT-PIV gas velocity measurements in a regular body-centred-cubic packing of glass balls.

2.1 Motivation

Martins et. al. (2018a, b) showed that RT-PIV measurements allow for the correction of strong optical distortions for synthetic setups and real PIV measurements behind single sphere arrangements. In the present work, we consider the application of the technique to more complex systems like the bcc packing described in Section 3, representing an idealised packed bed. According to Martins et. al., the recovered field of view after the distortion correction is 94,86% for two but only 5.20% for three d = 50mm spheres, aligned in front of the viewing plane. This information can be used for the design of the packing e.g. for the number of spheres that should be used and the set-up of the measurements. To clarify the effect of a larger number of spheres and also of the distance between the spheres in a bcc packing, the experiments by Martins et al. were extended.

2.2 Experimental setup

In our reconstruction of the experimental setup by Martins et al. (2018a, b), only minor changes concerning a more precise alignment were carried out. The basic principle of a target and the spheres aligned to the centre point of the camera sensor (Imager LX 8M from LaVision with a 60mm Nikon Nikkor Micro lens) remains the same. Therefore, as can be seen in the experimental setup shown in Fig. 1, a checkerboard pattern in a frame and the sphere holder are connected to three manual linear traversing stages to generate a 3D motion. For a linear alignment and the possibility to vary the distance between camera, sphere holder and target, the components are placed with carriers on a rail.

The 3D motion of each component allows for their alignment so that a small cross on the checkerboard pattern is moved until it appears at the centre of the image. Then it is assumed that the centre point of the camera sensor and the centre of the spheres are along the optical axis and therefore not affected by optical distortions.

One to four spheres with 40 and 50mm diameter are placed on a carrier on the rail, which allows for varying distances between the spheres (0mm, 3mm, 6mm and 9mm). The first sphere touching the checkerboard is aligned with the optical axis by moving the sphere holder in a way that the calibration marker on the background appears centred in the image and thus, in the centre of the sensor. Afterwards, the other spheres are placed on additional carriers on the same rail as the first sphere to ensure a good alignment along the optical axis, defined by the first sphere and the camera sensor. Different distances between the spheres can be realised by moving the carriers. During image acquisition, the scene is illuminated from the back with a white LED panel.

The obtained images are corrected using a ray tracing approach similar to Martins et al. (2018a, b), which is described in more detail in Section 4.2. The main difference between the correction of single sphere arrangements and the more complex packed bed application, described there, is the calibration used. In the single sphere case, only internal camera parameters, such as lens distortions, are considered, while for the complex setup external parameters, such as global camera position and rotation, are also calibrated.

2.3 Results

In Fig. 2 the acquired (Fig. 2a), simulated (Fig. 2b) and corrected image (Fig. 2c) for two aligned d = 50mm spheres touching each other (0mm distance) are shown. The simulated image is generated with the ray tracing step, using the known geometrical and experimental data of the set-up. It provides the relationship between distorted and corrected pixel positions and is used to generate the recovered image.

The method is applied to different test cases varying the sphere diameter from d = 50mm to d = 40mm and applying different distances between the spheres. Table 1 gives an overview of the most important parameters of each test case, including the amount of spheres, diameters, distances between the spheres and the distance between the camera sensor and the checkerboard target. The table also provides the recovered field of view (FOV) inside the blocked region defined by the ratio of the recovered area inside the sphere and the entire area behind the sphere. The areas are calculated using pixel based diameters of the circle belonging to each area. The circles are marked exemplarily in Fig. 3, with blue indicating the sphere and red the recovered area. The figure shows the corrected images for two spheres with different distances. The data from Martins et al. (2018a, b) is also included into Table 1 (M1 and M2) but the FOV is recalculated based on the method proposed here for better comparability.

Table 1

Parameters for simple sphere arrangements including the recovered FOV. The cases M1 and M2 belong to the data published by Martins et al. (2018a, b).
Sphere diameter	Distance from camera sensor	Distance between spheres	Number of spheres	Recovered FOV [%]	Case No.
50mm	500mm	0mm	2	94,86	M1
	500mm		3	5.20	M2
	693,5mm		1	3,27	0
			2	96,87	1
			3	8,90	2
40mm			1	3,03	3
			2	93,14	4
			3	9,16	5
		3mm	2	73,61	6
		3mm	3	7,76	7
		6mm	2	56,43	8
		6mm	3	5,98	9
		9mm	2	43,83	10
		9mm	3	4,04	11

Comparing the d = 50mm test cases M1 and M2 with the repeated ones (1 and 2), a slightly larger recovered field of view is obtained, which can be explained by changes in the experimental setup, like the used camera and its arrangement. The larger distance of the camera was necessary to generate the depth of field that is required for the correction behind four spheres. The four sphere result is not included in Table 1 because of alignment issues which will be discussed in the following. Besides this, it is very challenging to obtain the required depth of field in the experimental setup, especially in terms of the acquisition of particle field images. This will be discussed in more detail in Section 5.1.

Four aligned spheres are of interest because the recovered FOV is less than 10% behind one or three spheres. On the other hand, the arrangement of two spheres yields the highest recovered FOV, even if one considers that a distance between the spheres will lead to an immense decrease as shown in Table 1 and Fig. 3 for two d = 40mm spheres with a distance of 3mm (Fig. 3a), 6mm (Fig. 3b) and 9mm (Fig. 3c).

It is assumed that the application of the method behind an even number of spheres yields the largest possible recovered FOV. In this case, most distortions introduced by one sphere are cancelled by the subsequent one, as shown in Fig. 5, since exitant and incident ray directions are similar in the central regions of the spheres and these have an identical refractive index. This is established experimentally by the evaluation of the four sphere arrangement shown in Fig. 6.

Already on the raw image for the four sphere case (Fig. 6a), it is apparent that a perfect alignment of the spheres was no longer possible and the distortion pattern inside the central region is lacking the expected symmetry. Also the alignment cross does not appear in the centre of the image anymore. For better visibility a red cross is added to the image, marking the point where it is expected, if perfect alignment along the optical axis would have been achieved. This misalignment might be caused by tolerances in the assembly. Here, the most probable explanation is a slight rotation far below 1° of the rail where the sphere carrier sits. In combination with the increased distance to the optical axis when placing more spheres and the magnification effect of the balls, which act as lenses, any minute error is amplified. The consequences of the misalignment were not completely captured by the computer generated image (Fig. 6b), so that distortion remains in the corrected image (Fig. 6c). Furthermore, regions close to the rim appear blurred. Nevertheless, the results clearly establish that almost the entire region behind the spheres can be recovered, as expected for a setup with an even number of spheres. Because of the remaining distortion, a reconstructed FOV for this case has not been calculated. However, it is expected to be close to the two sphere case without distance between the spheres, if the aforementioned remaining issues are solved.

The information derived from the simple setup described above, was then used for the design of the setup and measurements of the idealised packed bed described in the following.

3.1 Packing

The packed bed is represented by a bcc packing consisting of d = 40mm high precision N-BK7 glass balls in the measurement zone, restoring optical access, as well as polypropylene precision balls in the remaining regions. The setup is a modular system where 16 layers of spheres in bcc arrangement with 3 x 3 spheres in a full, odd layer and 2 x 2 spheres in a weak, even layer are connected with a melting adhesive. Because of this, a small bridge between each sphere exists, leading to very small deviations from the theoretical bcc geometry. This unknown factor in terms of geometry will influence the ray tracing based reconstruction and is discussed in more detail in Section 5.3.

With additional modules consisting of a weak and a full layer, a bed height with up to 21 layers can be generated, while for the measurement region one of the modules is replaced by individual glass spheres. The polypropylene spheres in the measurement zone are covered with a frosted black colour to reduce reflections.

As discussed in Section 2.3 the distance between spheres has a large influence on the size of the recovered FOV. For a bcc packing with d = 40mm spheres, the theoretical distance between two spheres is 6.188mm and thus close to the cases with 6mm distance between the spheres shown in Table 1. The reconstruction of the view through one and three spheres only yields a small recovered FOV. Also for two spheres, the optimal recovered FOV is reduced from 93,14% to 56,43% through the distance of 6mm between the spheres. This provides, nonetheless, a large region where a velocity vector field can be obtained and the measurement plane for RT-PIV in the aforementioned packed bed has therefore been chosen between the second and third sphere, as seen from the camera.

3.2 Bulk reactor

The bcc packing is placed inside an acrylic glass reactor designed to have symmetric conditions in terms of geometry and inlet flow. By combining both properties also the flow inside and above the packing is assumed to be symmetric. To generate a homogenous inlet flow over the entire cross section, a diffuser in combination with a honeycomb and a random packing of d = 4mm glass spheres is used. It is homogenised by a second honeycomb and three further layers of spheres before entering the bcc packing through a ground plate. A hole pattern distributed over the ground plate with d = 4mm holes leads to a defined gas flow entering the packing. To reduce wall effects, this pattern is omitted in the rim region of the spheres. Bearings embedded in the ground plate hold the d = 40mm spheres of the packing in place and guarantee together with the walls of the reactor the correct positioning of the bcc packing. The acrylic glass walls possess an extended outlet zone to minimise the influence of the ambient conditions. Pressurised air controlled by a Bronkhorst Mass-Stream Controller (D-6371; 500 l/min air) is entering the reactor through the T-device of the diffuser together with Di-Ethyl-Hexyl-Sebacat (DEHS) as liquid PIV tracer particles with a particle size below 10µm, which are provided by a liquid nebulizer Type AGF 10.0 from Palas GmbH. The flow scheme of the gas flow is provided in Fig. 7.

3.3 Setup for RT-PIV

The experimental setup is shown in Fig. 8. The bulk reactor is placed on an Isel 3D-traversing system allowing for a constant distance between camera (Imager LX 8M from LaVision with an 35mm Nikon Nikkor lens) and the laser light sheet generated by a Quantel Q-smart Twins 850 Nd:YAG PIV-Laser, which was operated with a wavelength of 532nm, maximum energy of 380mJ/pulse and a frequency of 2.5Hz and controlled by the PIV software Davis 8.4. A programmable timing unit is used to synchronise laser and camera. By keeping the distance constant and moving only the reactor, the calibration effort is reduced.

In addition to the distance between camera sensor and light sheet, here 530.9mm, also the orientation between camera and packing as well as the focal distance and camera lens distortion parameters have to be known for the ray tracing based distortion correction step. Therefore, a two-step calibration is performed with two different calibration targets: first, a commercial 3D-target from LaVision (Type106-10), placed in the measurement plane for pinhole calibration in the PIV software Davis 8.4. The most important parameter gained from this step is the focal length of a pinhole camera approximation of the camera system, which is different from the nominal focal length of the real lens that is used. The second target, called ray tracing reference target (RTRT), is shown partly in Fig. 9.

The target consists of a checkerboard pattern with small cross marks in the centre of each checkerboard; these are aligned with the theoretical position of the centre of each sphere in the bcc packing. ArUco markers are placed in the gaps in between. The target is fixed to the back wall of the reactor assuming that the orientation of the target thus is the same as those of the reactor and the packing inside.

For calibration, the target is imaged at the laser light sheet position as well as its position during velocity field measurements (by moving the whole reactor with the traversing system). The gained information is used for three different tasks: (i) for pose estimation, which delivers the orientation of the camera with respect to the packing, and to obtain the distortion parameters needed for the ray tracing based reconstruction; (ii) to validate the ray tracing after simulating the scene seen through three spheres, as explained in more detail in Section 6.1, (iii) to determine the exact position of the measurement region.

The horizontal origin (x-axis) is chosen to be the centre of the reactor while the vertical axis (y-axis) provides information about the bed height above the ground plate. The depth (z-axis) is described by the camera viewing direction, with its origin at the inner side of the front acrylic wall (Fig. 8). The RTRT is back-illuminated with a green LED to provide a wavelength corresponding to that used during PIV recording because of the dependency of the refractive index on the wavelength. The refractive index is an important parameter for the ray tracing simulation and thus of great importance for the validation as discussed in Section 5.2.

4.1 Distortion correction

The distortions that are introduced through the transparent geometry are corrected using two optical ray tracing simulations of the light propagation through the experimental setup (Fig. 10). In the first pass, the image acquired in the experiment is considered as a light source. The light is then traced back to its origin in the packing by exploiting the reversibility of light transport. This yields an approximation to the light field, that is the distribution of light energy, on the investigation plane (light sheet) at the moment when the image was taken. In the second pass, we compute the image that would have been obtained without the transparent spheres (Fig. 10 right). This is the corrected particle image and used for further PIV processing.

The detailed reconstruction steps are as follows. First a 3D computer model of the experimental apparatus is generated. In our case this was directly available from the CAD files of the bulk reactor that were used for manufacturing it. Discrepancies between the computer model and the real experiment are thereby currently ignored. To account for the complex camera system, which would be too difficult to model explicitly, we use a pinhole camera approximation of it with lens distortion that is fitted using an intrinsic and extrinsic camera calibration. The fitting is already implemented in the LaVision DaVis Software as well as the OpenCV library (Bradski, 2000). Since the intrinsic parameters only depend on the camera that is used, it is carried out in an offline step prior to a measurement series. For the extrinsic calibration, the ArUco markers of the RTRT are used, which are directly available for recognition in the OpenCV library (Bradski 2000). Once the camera calibration is complete, the experimentally captured image is used as source and ray tracing is performed from the pinhole approximation of the camera. This yields the original light field distribution on the PIV investigation field (Fig. 10 left). The light field is then the input for a second ray tracing pass with a computer model of the experimental apparatus where the distortion-causing transparent spheres have been removed (Fig. 10 right). This results in a distortion-free imaging of the tracer field that can be used as input for standard PIV software. A modified version of the open source software PBRT (Pharr and Humphreys 2010) is used in our implementation, which was extended to provide complete information of the light paths in the experimental apparatus.

Before correcting the distorted particle field images obtained in the experiment (Fig. 11 left), a time filter, based on subtracting the minimum value for each pixel of a given number of consecutive images, is applied to remove the image background as well as minimise the effect of reflections (Fig. 11 middle). This helps us to avoid artefacts appearing in the corrected particle field images (Fig. 11 right).

4.2 PIV evaluation

The corrected images are re-imported to DaVis 8.4 as double frame images and attributes like the time step between two PIV-frames and each double image as well as the proper scaling are applied. A second time filter is carried out to remove artefacts which might appear during the ray tracing correction step.

The PIV evaluation then uses a standard cross-correlation method as a multipass calculation with a decreasing interrogation window size from 64x64 pixels to 32x32 pixels. Before calculating the vector field from the particle field images, masks are applied to remove regions where no signal can be obtained (see Section 5.2) or no reconstruction is possible in the rim regions of the spheres.

After vector field calculation, standard PIV post processing steps are carried out like filtering non-valid vectors and calculating required data like average images, standard deviations, etc.

5.1 Image quality and depth of field

To generate reliable results with the RT-PIV method described above, a high fidelity image has to be captured in the experimental apparatus. In terms of particle field images, the plane illuminated by the light sheet has to be in focus and have good contrast. In this study, the acquired particle field images for RT-PIV consist of two parts (Fig. 11). One region where no additional treatment for generation of optical access is necessary as the surface flow and the interstices between two rows of spheres in the surface layers, and a second region behind the spheres, where optical access is generated by the use of the transparent balls. These transparent spheres act as lenses and shift the focal plane from the measurement plane illuminated by the light sheet (green line in Fig. 12) to a plane towards the camera (black dotted line in Fig. 12). To record focused particle field images from both planes, the one in the surface flow above the bed and the one behind the transparent spheres, both have to be in focus. Furthermore, the RTRT on the back wall of the reactor (grey rectangle in Fig. 12), illuminated with an LED, also has to be in focus. This leads to the requirement of a very large depth of field, bigger than the reactor depth, for successful application of RT-PIV.

The required depth of field can be generated by increasing the distance of the camera to the plane which has to be in focus or by closing the aperture. The latter will lead to a loss of light reaching the camera sensor, resulting in a lower image quality. To avoid this, the distance was increased first. For a small aperture, which is recommended for imaging small tracer particles in PIV applications (Raffel et al. 2018), a distance of 1m should be enough to generate a depth of field of more than 200mm based on the lens that was used. Considering that the distance between the front wall and the RTRT is around 142.38mm, a depth of field of 200mm should result in focused particle field images. However, even by testing different distances and closing the aperture further, it was not possible to get all three planes in focus. This suggests that the focal plane created by the transparent spheres is shifted with increasing distance of the camera. This makes it impossible to have all three planes in focus by increasing the distance between camera and reactor.

In consequence, the camera was placed as close as possible to the bulk reactor and the aperture closed to still obtain a reasonable measurement signal. As the best compromise between the required depth of field and the tracer particle intensities, the camera was placed at a distance of 530.9mm and the aperture closed to f-16 (further closing would lead to intensities too close to the camera noise). Already Martins et al. (2018a, b) encountered this issue and mentioned that a slight blur in the background has to be accepted as a compromise between the different focal planes that are required.

The current measurements yield particle field images with intensities of around 100 to 200 counts for the tracer particles, compared to maximum 4096 counts in the 12-bit mode of the camera. For further enhancement, the use of a different lens that allows for a larger depth of field with bigger aperture for equal distances and with similar or even higher magnification can be considered. Other alternatives are higher laser energy density or brighter tracer particles.

The laser used for creating the light sheet is already operated at its maximum energy of 380mJ/pulse. Thus, the intensity of the tracer particle images cannot be enhanced easily. Also an increase of the energy density by illuminating a smaller area does not allow for a significant improvement.

Working with fluorescent particles to deal with reflections of the laser light inside the packing would be desirable. DEHS was chosen as a tracer particle because of the short degradation time and low toxicity. Most fluorescent particles cannot remain in the surrounding and have to be removed with a suction device, influencing the measurement conditions. Furthermore, the fluorescent signal has a much lower intensity than the signal from Mie scattering, amplifying the aforementioned problem with the illumination intensity.

5.2 Illumination and recovered area

For the illumination not only the intensity and the closed aperture are issues, also the access inside the packed bed is a limiting factor in terms of illumination. The current measurements are carried out above and in the top surface layer of the variable height bed. Already there, the issue caused by the illumination with a light sheet can be seen clearly, as shown in the particle fields in Fig. 11. In particular, a region without measurement signal in the upper part of the centred row of spheres (left spheres seen in the images) is apparent. With respect to the most pronounced distortion effect of the spheres, which is the mirroring, this region can be explained by the shadow of the underlying weak layer of spheres blocking the light sheet. The area belonging to the shadow is marked with red in the different particle field images in Fig. 11 before and after applying the ray tracing correction. Because of the sphere distances in the bcc geometry, there is already a loss of recovered area and this is further limited by regions without measurement signal. For measurements inside the packing, also the overlying layer of spheres will cause the same effect and only a small recovered area will remain.

The light sheet hitting the underlying weak layer also leads to strong reflections, marked green in Fig. 11, that have to be handled by the application of filters and masks to ensure correct PIV evaluation. These masks further reduce the area where a vector field can be recovered.

To handle the illumination issue, alternative illumination techniques can be considered, for example a light source inside the packing or volumetric illumination by LEDs from the outside. Until now, no satisfactory solution to these issues has been found. One possible approach would be to use small lasers with a size approximately of the sphere diameter. However, they do not have sufficient power to obtain sufficient intensities on the tracer particle field images. Another alternative is to illuminate the apparatus with LEDs. This, however, will cause strong reflections and the usage of fluorescent dye is indispensable. Furthermore, not enough energy will be available to generate the required depth of field and a good image quality.

5.3 Parameters for the ray tracing based reconstruction

To create a simulated 3D scene from the related measurement position, geometric parameters, distances in the setup and material properties like the refractive index have to be known. All these parameters represent potential error sources for the ray tracing correction step. For example, an inhomogeneity in the material of the glass balls will lead to a varying refractive index and distortion of light rays that is not taken into account in the simulation. The refractive index for the 40mm spheres is considered here to be n = 1.530, which is based on numerical experiments. For the numerical determination of the refractive index, the single sphere setup, described in Section 2.2, was simulated and the refractive index varied to match the visible pattern.

Another important parameter is geometric alignment of the spheres. In the simple sphere arrangement discussed in Section 2.2, it was possible to move the spheres and the camera towards each other to obtain an almost perfect alignment. This approach is impractical in a packing and the geometry has to be treated as is with deviations from the ideal bcc packing. The observed deviations can be explained by errors in the manufacturing process, e.g. a ± 2.5µm tolerance for the d = 40mm sphere diameter as well as the tolerances for the bulk reactor itself. These effects accumulate and lead to an approximately 3mm reduced bed height for the 21 layers in our setup compared to the theoretical bed height for the bcc packing when assembling spheres manually in the reactor. Fixing the spheres to a modular system does not solve the problem because then additional material between the contact points of the spheres is required. These approaches lead, in our case, to a 4mm higher bed with 21 layers compared to the theoretical bcc packing. Furthermore, the offset is not equally distributed but varies between layers. This inevitable misalignment of the spheres compared to the theory is visible in the images used for the ray tracing validation discussed in the following Section 6.1. to partially compensate for this geometric discrepancy, an offset is manually applied to the sphere positions in the ray tracing simulation.

Martins et al. (2018a, b) showed that ray tracing can be used for the correction of strong distortions caused by transparent geometries for artificial images as well as real particle fields. Because of the more complex geometry and more challenging circumstances of the measurements in a packed bed, a validation of the method was carried out. This can be done by a comparison of the acquired RTRT images with the computer generated ones and also based on the obtained vector fields by comparing the distorted region behind the spheres and the free surface flow where no correction was necessary.

6.1 Ray tracing validation

To validate the computer model of the physical setup, we consider the calibration images with the RTRT in the background and the laser sheet in focus. By simulating the image formation, we can determine the alignment of the geometry as well as the accuracy of the calibration by comparing the simulated and the experimental calibration images. This is repeated for every measurement position due to propagating errors as described in Section 5.3.

In the simulation, we do not consider depth of field or production tolerances which appear in the measured images. Furthermore, the simulation lacks camera noise and the texture introduced by the light diffuser as well as the reflections on the reactor wall. Finally, we also do not simulate the exact luminescence of the lighting. All of this makes the use of difference-images impractical, since they would show all of these effects as errors. We therefore compare simulated and calibration images visually, as in Fig. 13 and Fig. 14.

For a successful calibration, we expect to see a sub-pixel accuracy in the background of the images (red rectangle in Fig. 13), meaning that the checkerboards and ArUco markers should not deviate more than one pixel from each other. If we lay the images above each other we see that the background only differs by sharpness and texture, which is due to the depth of field issues discussed in Section 5.1 Since the corner positions of the checkerboard are identical, we can deduce that the calibration of both, internal and external camera parameters, was accurate. If we compare the view through the spheres (green and blue rectangles in Fig. 13), we see a similar pattern, again differing in sharpness since no depth of field is simulated. However, the corner positions of the checkerboards, which are visible through the spheres and indicated by red lines in Fig. 13, show a clear deviation. In the case of the lower right sphere, i.e. the sphere on the optical axis (green rectangle), we see a deviation of more than six pixels. In the case of the off-axis sphere, i.e. the lower left on (blue rectangle), an even stronger deviation of up to 31 pixels can be observed.

These are substantial errors and they result from a misalignment of the geometry between physical and simulated setup. Removing them completely is very challenging as the system is highly sensitive to differences in the geometry. However, with respect to the PIV reconstruction, the errors are not as severe as they appear at first sight. Because of the magnification through the three layers of spheres for the RTRT, the misalignment errors are also magnified. As the size of the checkerboards is known, one can compute the approximate offsets on the background plate. These values are also given in Fig. 13 and Fig. 14. The results show that the error through the spheres given in physical units (mm) is also larger than in the background. However, instead of a factor of 6.7 and 31.3 between background and sphere centre for the pixel values, we get, due to the magnification, factors of 1.7 and 8.0 for the dimensions on the background plane, respectively. The worst deviation in the above shown images is 0.8mm between the experimental and the simulated calibration target image.

The deviations depend on the number of layers inside the bed, as can be seen in Fig. 14 for a 19 layer bed. There, the differences of the patterns inside spheres (blue and green) differ from what is shown on Fig. 13. For this reason, it is absolutely mandatory to adjust the offset in the simulation for each layer individually, as described in Section 5.3.

The aforementioned deviations are with respect to the RTRT on the reactor back wall, which is seen through three layers of spheres. However, the stability of the system increases if we use an even number of spheres, as discussed in Section 2.3. Since we reconstruct the particle fields behind two layers instead of three, we expect to see better agreements than mentioned above.

The here proposed calculation of uncertainties relies on the identification of corners and measurement of known lengths. This is only robustly possible in regions with relatively constant distortion, limiting it to the centre regions of the spheres. Other methods for quantifying the error and based on the complete image matrix are also used for validation, but are, due to their numerical complexity, beyond the scope of this paper and will soon be published separately.

Errors are amplified when approaching the rim regions of the spheres, as the distortion increases rapidly. Visually comparing layer 17 and 19, the computer generated image for layer 17 seems to match much better the computer generated one than in the layer 19 case. This can be explained by an increasingly difficult geometric alignment with increasing height, since each layer adds geometric uncertainty. To validate both aforementioned effects, the reconstruction behind two spheres, as well as the differences of the ray tracing validation images in different layers, we consider the velocity vector fields.

6.2 Velocity vector fields

The second validation method for the ray tracing based correction considers the obtained vector fields after PIV evaluation. For the measurements at the surface of the packed bed, which is considered in the following, all vector fields contain two different regions, one where the ray tracing based correction is indispensable, since the region is behind the transparent spheres, and a second region, above the packed bed, where PIV measurements can be carried out easily without any additional distortion correction. This region consists of the free surface flow as well as the interstices in between two rows of spheres where no sphere is situated in the line of sight. During ray tracing correction, both regions are treated in the same way. As shown in Fig. 15, no significant differences can be observed if one compares the velocity fields in the easily accessible region above the bed obtained from the raw particle images (Fig. 15, top left) and the ray tracing corrected ones (Fig. 15, top right). Based on the difference of both images normalised with the maximum velocity in the surface flow (Fig. 15 bottom left), a mean difference value of 0.039% is calculated, while the maximum difference is below 3.1% in the region close to the masks applied during PIV evaluation in the sphere rim region.

Further, a successful application of the ray tracing method should result in flow fields that match at the boundary between distorted and undistorted regions. This is depicted in Fig. 15, bottom right, which shows a zoom to the region between two spheres, as marked with a green rectangle in Fig. 15, top right. A continuous and steady course of the velocity field is obvious in the figure, except for regions where reconstruction was not possible and masks had to be applied for the PIV evaluation. In the interstice between the spheres, a vortex can be observed, which is well resolved even though parts of it are situated behind a sphere. This shows that the distortions induced by the transparent spheres were corrected successfully.

In Section 6.1 we discussed that errors close to the rim regions of the spheres are more pronounced as the number of layers increases. The ray tracing validation hence showed a larger deviation between acquired and computer generated images there. It was explained that this issue is less significant in the measurement plane behind two spheres, compared to the three spheres in front of the RTRT (Section 2.3). This can now further be supported by checking the vector fields for different bed heights (e.g. 17 or 19 layers, Figs. 15 and 16 respectively). The vector fields for layer 19 (Fig. 16) show similarly good results in terms of continuous and steady course of the flow field, as already discussed for layer 17 in Fig. 15 before. The recirculation zone between the two rows of spheres is well and continuously resolved. The corrected region behind the spheres joins very well the undistorted surface flow field.

To support the finding of a continuous and steady flow field and a good agreement of the surface flow and the corrected region behind the spheres in different bed heights (layer 17 and 19), vertical velocity profiles for the vertical velocity component are provided in Fig. 17. Here, we focus on a region inside the green marked boundaries between x = 17.5 ... 20.7mm. While the lower limit (x = 17.5mm) represents a region where mainly ray tracing correction was necessary, the upper limit (x = 20.7mm) is completely in the interstices between two spheres, where no reconstruction has to be applied. All profiles show a continuous and steady course. For x = 17.5mm this course is interrupted at a bed height of about 398 and 445mm, for 17 and 19 layers respectively, because of the mask necessary during PIV processing and not allowing the vector calculation at this place. The mask is applied to disable regions without measurement signal or regions where no reconstruction was possible, as it is the case for the rim regions of the spheres where strong distortions appear.

A detailed discussion of the measured flow fields and the variation of parameters like the particle Reynolds number and bed height is beyond the scope of this article and will be published separately soon.

The validation was carried out based on a small area of the entire vector field which we were able to reach due to the perspective applied to the left row of spheres in all shown images. This perspective allows for the reconstruction of the information in the rim regions of the area behind the spheres. For an extension of this method and a complete validation it is planned to perform RT-PIV in a multi-camera setup. This allows for different perspectives to the measurement plane to recover the complete rim regions. With this, a better comparison of the flow field surrounding the spheres and behind these will be possible.

In the present work RT-PIV is applied to a complex measurement test case represented by a bcc packing of transparent spheres. Therefore, firstly the work of Martins et. al. was extended to obtain essential information for an optimal experimental design. By investigation of up to four spheres it was found that the view through an even number of spheres yields an almost complete recovered FOV while odd numbers will only lead to a small recovered area. It was also found that the distance between the spheres is important and the recovered FOV decreases with increasing distance.

Based on this information, RT-PIV was applied to a packed bed designed to obtain optimal measurement conditions. During the realisation of the experiments, difficulties like image quality, illumination and alignment issues appeared, which were discussed in detail in the presented work.

Nevertheless, the application of RT-PIV was successfully validated based on a comparison of a computer generated and an acquired image of a reference target, establishing that the light transport simulation and the image correction are in good agreement. Furthermore, the corrected velocity vector fields of a gas flow through the packed bed were used for an intrinsic validation. By comparing regions where strong distortions introduced by the glass balls had to be corrected and regions with a direct line of sight, velocity fields in good agreement were obtained. This established the successful application of the correction method.

In the future, the RT-PIV method will be used to evaluate the velocity field through a packed bed with different operational parameters, e.g. Reynolds number, and geometrical positions. The aim is to obtain detailed gas velocity validation data for numerical simulations of packed beds.

Acknowledgments

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 422037413 – TRR 287.

Péter Kováts and our student Zahin Thamed is acknowledged for his help during experiments.

Ethical approval

Not applicable.

Competing interests

None.

Authors’ contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by Christin Velten, Mirko Ebert, Christian Lessig and Katharina Zähringer. The first draft of the manuscript was written by Christin Velten and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 422037413 – TRR 287.

Availability of data and materials

For access to the used datasets please contact the corresponding author ([email protected]).

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No competing interests reported.

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Ray tracing Particle Image Velocimetry – Challenges in the application to a packed bed

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Abstract

Figures

1. Introduction

2. Distortion Effects Of Single Spheres Arrangements

2.1 Motivation

2.2 Experimental setup

2.3 Results

3. Design Of The Experimental Setup

3.1 Packing

3.2 Bulk reactor

3.3 Setup for RT-PIV

4. Post Processing

4.1 Distortion correction

4.2 PIV evaluation

5. Challenges

5.1 Image quality and depth of field

5.2 Illumination and recovered area

5.3 Parameters for the ray tracing based reconstruction

6. Validation

6.1 Ray tracing validation

6.2 Velocity vector fields

7. Conclusions

Declarations

References

Additional Declarations

Supplementary Files

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