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Some concept of Optical Measurement Techniques in Thermal Sciences are Absorption Techniques, Alternative Approaches, Calibration Details, Computerized Tomography, Convolution Backprojection. Main points of this lecture are: Crystal Growth, Shadowgraph, Schlieren, Crystal Growth, Crystal Size, Experimental Data, Grown Crystals, Convection, Currents, Growth Chamber
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Effect of Crystal Size Competing Mechanisims in Concvection Tomographic Reconstruction of Schlieren Data Convection Around a Growing KDP Crystal Growth Patterns in the Diffusion Regime
In the previous sections, it is seen that the crystal growth rate is positively correlated with the local concentration gradients. The gradients in turn depend on the size of the crystal, ramp rate and the rate of rotation. It is of interest to know if the gradients are explicit functions of time. An absence of time-dependence would show the process to be quasi-steady, and the results obtained in the present work would be applicable with greater generality. To test the experimental data for the presence of growth history and inertia effects, the following experiments were performed: Previously grown crystals of small, medium and large sizes were suddenly inserted into the supersaturated solution. The solution was subjected to cooling at the prescribed ramp rate (of ). The short time convection currents around a passive crystal have been compared with those that evolve gradually around a growing one when the growth process takes place from a seed, for a longer duration of time (~ 60-70 hours). A favorable comparison in terms of convection patterns is an indicator that the concentration gradients respond directly to the process parameters, and are not explicit functions of evolutionary time.
Figure 5.25 shows instantaneous schlieren images of buoyancy-driven convection around the crystal of varying sizes (small, medium and large) over a duration of 3 hours. Compared to the time durations generally required, this is a small time scale. Figure 5.25 shows that short time transients are indeed present in the solution, but they decay within a time frame of three hours for the three sizes considered. Over this time frame, the change in the crystal size was found to be negligible. Figure 5.25 shows that the strength of convection is weaker during the transient phase, and is the most vigorous when steady state appropriate for the size of the crystal is reached. The convection plumes during the transient phase can be unsteady and unsymmetrical. The loss of symmetry is related to the imperfections in the initial crystal geometry. The unsymmetrical plume is responsible for inducing flow unsteadiness above the crystal. With the passage of time, minor imperfections of the crystal are accommodated and the plume becomes symmetric as well as steady. The schlieren images in Figure 5.25 recorded at the end of 3 hours of experimentation correlate well with those shown in Figure 5. (Lecture 31). The similarity of buoyant plumes in the two configurations reveals that the long-time crystal growth process is indeed quasi-steady. Figure 5.25 also shows that departure from quasi- steady growth conditions can result in loss of symmetry and unsteady plume movement, both of which are undesirable from the view-point of crystal quality.
The relative importance of buoyant convection over rotation is discussed in the present section in fundamental terms. Essentially, the increase in crystal size takes place owing to the deposition of excess salt in the solution on the crystal faces. The speed at which the deposition occurs is proportional to the gradients in solutal concentration prevailing in this region. The gradients in turn depend on the pattern of fluid motion in the crystal growth chamber. The influence of fluid motion on concentration gradients can be explained in terms of solutal boundary-layers ( diffusion boundary- layers ) adjacent to the crystal faces. The boundary-layer thickness decreases as the fluid velocity increases. Clearly vigorous convective motion will increase concentration gradients, ultimately leading to a rapid increase in crystal size.
The conditions under which flow is intensified can be understood in terms of the forces acting on the fluid particles. For the solution contained in the beaker, the external force arises from buoyancy (in a gravitational field), while inertia forces related to linear and rotational acceleration are also present. The motion, in all situations is retarded by fluid viscosity. The forces are conveniently explained in terms of the dimensionless parameters Ra, the Rayleigh number and Re, the Reynolds number. Rayleigh number denotes the ratio of energy available in a buoyancy field and energy required to overcome viscous forces. Reynolds number is a ratio of rotational and viscous forces.
Here, is acceleration due to gravity and is the volumetric expansion coefficient of the solution with respect to salt concentration. Further is the concentration difference between the saturated and super-saturated solution, the crystal thickness in the vertical direction, the kinematic viscosity and the mass diffusivity
. The symbol is the crystal rotational speed. In the present set of experiments, the linear dimension increases with time, till the excess solute in the vicinity of the crystal is fully utilized.
Imaging of three dimensional convection patterns around the growing KDP crystal is discussed here. Four view angles are considered for tomographic reconstruction of the concentration field. In the experiments conducted, the crystal is not disturbed during the recording of the projection data. In addition, longer durations of crystal growth have been considered. Since finite time is required to turn the beaker and record projections, experiments have been conducted when the concentration field around the crystal is nominally steady. Occasionally, the plume above the crystal is marginally unsteady, and a time-averaged sequence of schlieren images has been used for analysis. The increment in the view angles is 45 o^ , covering the range of 0 to 180^ o^.
The crystal growth process has been initiated by inserting a spontaneously crystallized KDP seed into its supersaturated solution at an average temperature of 35 o^. This step is followed by slow cooling of the aqueous solution. The cooling rate employed is 0.05 o^ C/hour. The seed thermally equilibrates with the solution in about 20 minutes. With the passage of time, the density differences within the solution are solely due to concentration differences. Adjacent to the crystal, the solute deposits on the crystal faces, and the solution goes from the supersaturated to the saturated state. When the crystal size is small, salt deposition from the solution on to the crystal occurs by gradient diffusion. Diffusive transport has been sustained for a longer duration of time (and larger crystal sizes as well) by maintaining a small degree of supersaturation in the solution.
The question of a correlation between the solutal distribution in the solution and the topography of the crystal can be partially addressed by reconstructing the solute concentration contours on select planes above the crystal. Reconstruction from schlieren images using principles of tomography is discussed in the present section. Unlike the hanging crystal configuration in Sections Comparision of Interferometry, Schlieren and Shadowgraph in a crystal Growth Experiment and Influence of Ramp Rate and Crystal Rotation on Convection Patterns, the crystal in the set of experiments was mounted over a platform. With this arrangement, the interference of the rising convection plume with the crystal holder is eliminated. Recording projections requires the apparatus to be turned; in this respect, what is collected is time-lapsed data. For meaningful analysis, it is thus necessary that the convection pattern be steady in time. Only those experiments where this requirement was fulfilled have been included in the discussion.
One parameter that defines the quality of the growing crystal is the symmetry of its faces. Since growth from an aqueous solution is governed by the strength and movement of buoyancy-driven convection currents, the convective field plays an important role in transporting solute from the bulk of the solution to the crystal faces. Hence, to ensure symmetric growth of the crystal, it is necessary to ensure that the convective field in the growth chamber retains a symmetry pattern. In the present discussion, a viewpoint that an axisymmetric convective field is favorable for crystal growth has been adopted. Thus, conditions under which the solutal distribution is axisymmetric have been delineated from the tomographic reconstructions.
Results are presented for experiments on crystal growth in the diffusion as well as steady convection regimes. In Section CONVECTION AROUND A GROWING KDP CRYSTAL , results have been presented in the form of a time sequence of schlieren images recorded at four view angles, corresponding to four pairs of optical windows of the growth chamber. The crystal is not disturbed during the growth process. In this respect, the measurement process is truly non-invasive, though with a drawback of allowing only a limited number of projections. The emphasis is towards understanding the diffusion-dominated and the stable growth regimes of the crystal. In both regimes, the concentration field is practically steady, particularly during the recording of the individual projections. The reconstruction of the concentration field over selected planes above the crystal and their relationship to the crystal geometry are discussed. The influence of crystal rotation on the convective field is not covered in the present discussion.
Figure 5.27 shows normalized concentration maps above the growing crystal as derived from the schlieren images of Figure 5.26. The value of represents a saturated solution, while the limit refers to the supersaturated solution at its instantaneous temperature in the growth chamber. The maximum crystal size at the end of each experiment has been used to non-dimensionalize the x and y coordinates. For hour, the concentration contours of Figure 5.27 in the bulk of the solution reveal an almost symmetric solute distribution. Symmetry can also be gauged by the identical nature of contour distributions for each view angle. The contours are not densely spaced, indicating low concentration gradients. The contours are localized in the vicinity of the crystal, with the bulk of the solution being at the supersaturated state. Hence this phase of the experiments corresponds to the growth process with transport across a diffusion boundary layer attached to the crystal. With the passage of time, the crystal size increases and the concentration gradients increase in strength. At this stage, the concentration field is accompanied by a partial loss of symmetry. Experiments of the present study revealed a marginal unsteadiness as well. This phase defines a transition process from diffusion-dominated transport of solute to the onset of convection. With the passage of time, a weak convection current is set up around the crystal. The images recorded at t =70 hours also show a greater loss of symmetry in the concentration contours and a denser spacing closer to the crystal are to be seen.
Figure 5.28 shows the variation of concentration with respect to the (vertical) y- coordinate in the crystal growth chamber. The concentration referred here is an average obtained in the viewing direction of the laser beam and across the image as well. For averaging the concentration values in the x- direction, namely normal to the light beam, 11 columns along the width were selected. The concentration profiles for t =1 hour (Figure 5.28(a)) overlap in the bulk of the solution above the crystal for all the four view angles, thus revealing a uniform distribution of solute concentration in this region. At t =35 hours (Figure 5.28(b)), the differences in the profiles are quite noticeable. These differences can be attributed to the temporary unsteadiness that sets in when a transition from diffusion- dominated growth takes place towards convection. With the passage of time, the profiles become increasingly symmetric and overlap with each other. The growth now takes place in the presence of an upward rising convective plume (Figure 5.28(c)). The symmetry in concentration distribution indicates an almost uniform and symmetric deposition of solute onto the crystal surfaces. It is to be noticed that for the three time instants, the profiles show differences in the vicinity of the crystal
. The morphology of the KDP crystal is pyramidal, as shown in Figure 5.29 in a photograph of a grown KDP crystal in the diffusion-dominated regime, at the end of 70 hours. The differences in the average concentration near the crystal can be traced to the non-symmetric shape of the crystal as well as large gradients at the crystal edges.
Crystal growth from a solution in which the limit of supersaturation has been reached is now considered. The density differences (ramp rate of /hour) as well as the crystal size (6 mm) are jointly large enough to sustain fluid motion in the beaker from the start. Experiments have been conducted under conditions of a steady plume rising from the crystal. Results for 30 hours of growth have been presented. For times greater than 30 hours, the plume became quite unsteady. Consequently, the projection data of individual view angles became uncorrelated.
Figure 5.31 shows the convective field in the form of schlieren images. A well-defined plume rising from the top surface of the crystal is seen at both times of t =2 and 30 hours for four view angles. At t =2 hours, the plume was temporally stable, though a swaying movement was seen from one projection angle to the other. The swaying motion was due to the disturbance of the flow field caused by the turning of the growth chamber for recording the projection data. The tilt in the plume at selected angles for small time is also related to the non-symmetric shape of the initial crystal itself. At a later time ( t =30 hours), the gradients are large enough to give rise to a stable and symmetric convective field. This is accompanied by uniform deposition of the solute on the crystal surfaces. As discussed in Section Growth Patterns in The Diffusion Regime, this regime yields the highest quality of the crystal in terms of symmetry of the faces and its transparency, along with the fastest growth rate.
Figure 5.34 shows reconstructed concentration profiles at selected horizontal planes above the growing crystal for the two time instants considered in Figure 5.31. The following observations can be recorded on the basis of the reconstruction.
