Depth Profile for Bulk Velocity and Solid-Volume Fraction

Considering that extensive amount of high-speed images are obtained for the discharge of each reservoir packing, only one flow from either the narrowest (W=6cm) or the widest (W=24cm) was analyzed in this work. The bulk velocity and solid-volume fraction depth profiles were examined at specific times t to examine _i

how each flow evolves in time. Since the images were captured at high frame rate (600PFS), the resulting PTV and hence bulk U and φ could fluctuate severely in time due to the small time interval. Thus, some local smoothing is required to

‘average-out’ these fluctuations for more meaningful bulk ‘instantaneous’ depth profiles and two methods were attempted. Firstly, we calculated four consecutive instantaneous depth profiles from t with a Δt=1/600-second interval and we used _i their mean to represent the bulk mean instantaneous properties att . For example, the _i velocity profiles att , _i t +Δt, _i t +2Δt, and _i t +3Δt are averaged to give U(_i t , y) and _i similarly for φ (t , y). In the second method, we extract sphere information at i t and _i

t +4Δt to calculate a new PTV giving instantaneous bulk U(i t , y) and φ(_i t , y) with an _i extended time interval of 4Δt .

For the flow from the narrowest packing (W=6cm), the two bulk instantaneous velocity depth profiles where compared at the three observation lines—from up- to down-stream—introduced in section 4.1 (see figure 4.1), in figure 4.7(a)-(c) at t = _i

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9,22, 35, 50, and 63 seconds. In each subplot, the mean instantaneous depth profiles are shown by the red line with the corresponding standard deviation at each flow height indicated. The blue dots show the instantaneous depth profile by using the extended time interval. The corresponding depth profiles for solid-volume fraction at the three guide locations are provided in figure 4.8(a)-(c). The U(t , y) and _i φ (t , y) i

from the thickest packing with W=24cm are examined at t = 35, 51, 65, 80, 93, and _i 109 seconds in figure 4.9 and 4.10, respectively, with each containing three subplots (a)-(c) for data obtained at the same three observation lines.

It is clear that the depth profiles from both estimation methods (the red line vs.

the blue dots) give nearly matching U and φ results for both the flows from the

narrowest and the widest packing over the inspected durations. However, severe deviation between the two depth profiles is observed for both U and φ from the two packing geometries—see U(t=9sec, y) in figures 4.7(a) and 4.7(c), φ (t=9 and 22 sec, y) in figure 4.8(c), U(t=35 and 51sec, y) in figures 4.9(c), and φ (t=35 and 51 sec, y) in figure 4.10(c). All these discrepancies are found near the free surface of each flow at early times and mostly at the reservoir gate or the guide exit.

Possible reason is attributed to mismatch spheres in the nearest neighbor method which in turn gives erroneous PTV data. Recall that a search circle of radius of one sphere diameter is employed to identify possible spheres in the consecutive

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image to be matched to the candidate in the previous image. We identify the sphere possessing the shortest distance from the candidate in the second image to obtain PTV result for the candidate. However, in the regime of rapid sphere motion, the candidate sphere can move a large displacement downstream leaving vacancy for other spheres—from upstream or from the bulk away from the wall—to fill in as illustrated at Time 4 in figure 4.11. These newly emerging spheres (in dark shade) are much closer to the candidate original location (in dashed circles) than the candidate later locations (in solid circles). It is also possible that the candidate sphere moves to the vicinity of the original position of other candidates—as marked by the bold circles.

All these scenarios can result in erroneous matching between two consecutive images giving wrong PTV results at Time 3. Such rapid change in sphere configuration may occur when the reservoir spheres are pushed through the gate at flow initiation like that shown in subplot (a) in figures 4.7 and 4.9. This scenario can also happen in loose flow at the free surface of the bulk leaving the guide exit in subplot(c) of those figures.

The occurrence of mismatching and hence incorrect PTV data becomes inevitable when too large a time interval is adopted—like that between Time 1 and Time 4 in figure 4.11 or that in the second evaluation scheme. Thus, the mean depth profiles from four instantaneous PTV (with Δt=1/600 second interval) should give us more accurate data than the calculation using 4Δt. In contrast, since the computation of φ

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only requires the instantaneous sphere configuration, the depth profile without time averaging should be more accurate.

9 seconds 22 seconds 35 seconds

50 seconds 63 seconds

9 seconds 22 seconds 35 seconds

50 seconds 63 seconds

(a) W=6 cm, at the gate

(b) W=6 cm, in the middle of the guide

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Figure 4.7 Instantaneous depth profile of bulk velocity for flow from the packing with W=6cm at (a) reservoir gate, (b) middle of guide, and (c) guide exit. With specific inspection times in each subplot.

9 seconds 22 seconds 35 seconds

50 seconds 63 seconds

9 seconds 22 seconds 35 seconds

50 seconds 63 seconds

(c) W=6 cm, at the guide exit

(a) W=6 cm, at the gate

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9 seconds 22 seconds 35 seconds

50 seconds 63 seconds

9 seconds 22 seconds 35 seconds

50 seconds 63 seconds

(b) W=6 cm, in the middle of the guide

(c) W=6 cm, at the guide exit

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Figure 4.8 Instantaneous depth profile of solid volume fraction for flow from the packing with W=6cm at (a) reservoir gate, (b) middle of guide, and (c) guide exit.

With specific inspection times in each subplot.

35 seconds 51 seconds 65 seconds

80 seconds 93 seconds 109 seconds

35 seconds 51 seconds 65 seconds

80 seconds 93 seconds 109 seconds

(a) W=24 cm, at the gate

(b) W=24 cm, in the middle of the guide

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Figure 4.9 Instantaneous depth profile of bulk velocity for flow from the packing with

W=24cm at (a) reservoir gate, (b) middle of guide, and (c) guide exit. With specific inspection times in each subplot.

35 seconds 51 seconds 65 seconds

80 seconds 93 seconds 109 seconds

35 seconds 51 seconds 65 seconds

80 seconds 93 seconds 109 seconds

(c) W=24 cm, at the guide exit

(a) W=24 cm, at the gate

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(c)

35 seconds 51 seconds 65 seconds

80 seconds 93 seconds 109 seconds

35 seconds 51 seconds 65 seconds

80 seconds 93 seconds 109 seconds

(b) W=24 cm, in the middle of the guide

(c) W=24 cm, at the guide exit

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Figure 4.10 Instantaneous depth profile of solid volume fraction for flow from the packing with W=24cm at (a) reservoir gate, (b) middle of guide, and (c) guide exit.

With specific inspection times in each subplot.

Figure 4.11 Illustration of the nearest neighbor method where mismatching spheres occur at Time 4.

Because the current PTV algorithm does not distinguish when such mismatch occur, we decided to use the instantaneous velocity from two consecutive images with 1/600 interval and the instantaneous sphere configuration to obtain U and φ in the following analysis.

We then estimate the bulk mass flow rate from lateral 2D images and from load cell signal that represents the total deposited weight. Using bulk solid fraction and velocity profiles from 2D images and assuming that the mass flux is constant across the chute, we may estimate the mass flow rate at a specific streamwise location by

0

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where ρ_s represents the sphere density, w represents the chute width and h is the local flow depth. The mass flow rate can also be measured from load cell signal as

dWdt

m^•LC = (4.3(b))

using the total deposited weight, W(t). Equation 4.3(b) is actually the time rate of change of W(t) in figure 4.2. The mass flow rates from equation 4.3(a) and 4.3(b) at different streamwise positions are compared in figure 4.12 for flows from different

reservoir packing widths. In these figures, the red squares are for m^•LC and the blue circle, dot, and star lines are for m^image.

(a) (b)

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(c) (d)

Figure 4.12 Instantaneous mass flow rate calculated by 2D image analysis (blue), and by load cell signal (red), with constant depth and different packing width (a) W=6cm, (b) W=12cm, (c) W=18cm and (d) W=24cm.

It is clear that m^image is overestimated throughout the observation for all packing width, W. We suspected that the overestimation is due to the error in volume fraction estimated from the lateral images. To estimate an upper bound for overestimation, we consider a 3D minimum volume fraction, φ_min =0.4, in which packing persistent contacts exists throughout the control volume. Nonzero φimage−φ_mincan result in

instantaneous difference in mass flow rate as



where φ_image, is the bulk volume fraction estimated by 2D lateral images. The results

obtained at the gate, middle, and the guide exit are shown by blue, red, and black respectively in figure 4.13 for the four packing widths. On the same plot, the

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overestimated flow rate from the load cell data, m^•image−m^•LC, are also shown by blue, red, and brown lines.

(a) (b)

(c) (d)

Figure 4.13 Instantaneous difference on mass flow rate, m^•image−m^•LCin solid lines and Δm^ in scattered points. With a 3.5 seconds interval and different colors stands for different streamwise positions. (a) W=6cm, (b) W=12cm, (c) W=18cm and (d) W=24cm.

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From figure 4.13, it is obvious that m^•image−m^•LCis smaller than the difference Δm from an assumed minimum • φmin in equation 4.3(c). This implies that the actual

flow is denser than φ_min. However, when the two data sets conform at later times, t>20 seconds for W=10 in figure 4.13(a) and t>40 seconds for the rest, the actual flow volume fraction asymptotes to φ_min and the bulk moves in a loose formation.

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