Granular flow through an aperture: influence of the packing fraction
For the last 50 years, the flow of a granular material through an aperture has been intensely studied in gravity-driven vertical systems (e.g. silos and hoppers). Nevertheless, in many industrial applications, grains are horizontally transported at constant velocity, lying on conveyor belts or floating on the surface of flowing liquids. Unlike fluid flows, that are controlled by the pressure, granular flow is not sensitive to the local pressure but rather to the local velocity of the grains at the outlet. We can also expect the flow rate to depend on the local density of the grains. Indeed, vertical systems are packed in dense configurations by gravity but, in contrast, in horizontal systems the density can take a large range of values, potentially very small, which may significantly alter the flow rate. In the present article, we study, for different initial packing fractions, the discharge through an orifice of monodisperse grains driven at constant velocity by a horizontal conveyor belt. We report how, during the discharge, the packing fraction is modified by the presence of the outlet and we analyze how changes in the packing fraction induce variations in the flow rate. We observe that variations of packing fraction do not affect the velocity of the grains at the outlet and, therefore, we establish that flow-rate variations are directly related to changes in the packing fraction.
pacs:45.70.-n, 45.70.Mg, 47.80.Jk
I Introduction and background
Because of its obvious practical relevance, the flow of granular media through an aperture has been intensely studied in the last years in vertical gravity-driven systems (e.g. silos and hoppers) Beverloo (); Kadanoff (); deGennes (); Trappe (); Jaeger (); Duran (); Ristow (); Nedderman (); Tuzun (); Savage (); Tighe (). The discharge of a silo through an orifice can present three regimes: a continuous flow, an intermittent flow, or a complete blockage due to arching Mankoc07 (); Mankoc2009 (); Ulissi ().
In the continuous flow regime, the mass flow rate (i.e. the mass flowing out per unit time ) is generally satisfactorily given by the so-called Beverloo’s law Beverloo (); BrownBook (): where is the diameter of the opening (assumed here to be circular), the bulk density of the granular sample, the acceleration due to gravity and the diameter of the granules whereas and are empirical, dimensionless, constants. The Beverloo’s law thus points out a value of the aperture size at which the flow rate is expected to vanish. Therefore, instead of , the effective aperture is to be considered. The value of has been found to be independent of the size of the grains and to take values ranging from 1 to 3 depending on the grains and container properties Nedderman1980 (). Nevertheless, some works Zhang (); Mankoc07 () claim that the only plausible value for is . It should also be noted that a recent work Sheldon () states that is just a fitting parameter with no clear physical meaning as the authors found clogging of the flow for apertures . In the jamming regime, the jamming probability has been shown to be controlled by the ratio of the aperture size to the grain diameter Mankoc2009 (); To (); Zuriguel03 (); Zuriguel05 (); Janda (); Ulissi ().
In many industrial applications, however, granular materials are transported horizontally, lying on conveyor belts desong03 () or floating on the surface of flowing liquids Guariguata2009 (); Guariguata2012 (); Lafond2013 (). In a two-dimensional (2D) configuration – or similarly for slit shaped apertures – one expects Beverloo’s law to be: BrownBook (). Recent works considered the discharge of a dense packing of disks driven through an aperture by a conveyor belt. For large apertures (), the flow-rate is continuous throughout the discharge. In this case, the number of discharged disks depends linearly on time and the flow rate (i.e. the number of disks flowing out per unit time ) obeys:
where and the constant reduces to the packing fraction Aguirre2010 (). Indeed, is the surface area of one disk so that is the number of grains per unit surface which, multiplied by the belt velocity and by the size of the aperture, gives an estimate of the number of disks flowing out per unit time. Note that Eq. (1) is equivalent to the 2D Beverloo’s law in which the typical velocity , understood as the typical velocity of the grains at the outlet, is replaced by the belt velocity . It predicts that the dimensionless flow rate is independent of and increases linearly with the dimensionless aperture-size . It is interesting to note that this empirical law was demonstrated to be valid for small apertures , even if the system is likely to jam and deviations from linearity might be expected Aguirre2010 (). Indeed, in 3D configurations, a marked deviation from the Beverloo’s scaling has been observed for very small apertures Mankoc07 (). Moreover, these previous works show that, unlike fluid flows, granular flows are not governed by the pressure, but rather controlled by the velocity of the grains at the outlet Aguirre2010 (); Aguirre2011 (). The latter does not necessarily depend on the stress conditions in the outlet region as proven by the experimental fact that, in gravity-driven systems, the typical velocity at the outlet is , independent of the pressure. These observations were corroborated in vertical gravity-driven systems Perge2012 ().
Even if the Beverloo’s law has been intensively discussed, the influence of the packing fraction, i.e. the ratio of the area occupied by grains over the total available area, has only been partially considered. However, it is expected that the flow rate can be altered by the packing fraction of the grains aside from their velocity. On the one hand, vertical granular systems are usually gravity packed in dense configurations, except in situations where inflow rate is controlled Huang2006 (); Huang2011 (), and little effect of the packing fraction is expected in usual conditions. But, on the other hand, in horizontal configurations the packing fraction can explore a large range of values and one can expect significant changes in the flow rate. Ahn et al studied granular flow rate in vertical silos filled under different conditions, which, as a consequence, lead to different values of packing fraction Ahn2008 (). However, aiming at relating flow-rate variations to changes in the pressure, they do not discuss the possibility that the variations could be due to changes in the packing fraction itself. In a more recent work, Janda et al studied velocity and packing fraction profiles at the outlet and they obtained a new expression, independent of , for the granular flow rate Janda2012 ().
In the present article, we study the discharge of monodisperse acrylic rings, driven through an orifice, at a constant velocity, by a horizontal conveyor belt. For various initial packing fractions, we report simultaneous measurements of the grains velocity, packing fraction and flow-rate throughout the discharge process.
Ii Setup and protocol
The experimental setup (Fig. 1) consists of a conveyor belt made of black paper (width cm, length cm) above which a confining cardboard frame (inner width cm, length cm) is maintained at a fixed position in the frame of the laboratory. A motor drives the belt at a constant velocity . The granular material is made of acrylic rings of thickness mm and external diameter mm.
Downstream, the confining frame exhibits, at the center, a sharp aperture of width . The aperture width can be tuned up to cm but we shall report data obtained for a single width cm. The aperture size is of about 10 times the grain diameter , so that the condition insuring the continuous flow, , is satisfied Aguirre2010 ().
The grains are imaged from top by means of a digital scanner (Canon, CanoScan LIDE200) placed upside down above the frame. In order to focus on the top of the grains without mechanical contact (gap of about 1 mm) and thus avoid friction between the grains and the scanner window, the latter has been replaced by a thinner one. The use of a scanner has the advantage of avoiding optical aberrations and makes it possible to obtain, for cheap, homogeneously lighted images with a high resolution ( pixels/mm, the grain diameter being thus of the order of 50 pixels).
Before the flow is started, the initial state of the system is obtained by placing inside the confining frame, in a disordered manner, grains which initially cover the surface area , where is the inner width of the frame ( cm) and, thus, the length in the flow direction that is initially covered with grains. We prepare systems with different initial packing fractions: ( cm); ( cm); ( cm) and ( cm). The homogeneity of the initial packing throughout the system is controlled by measuring the packing fraction along the flow direction in successive layers of width and thickness . Grains are locally rearranged if the packing fraction is not within of the chosen average .
The discharge is then initiated by setting the belt velocity to a chosen value. Experiments were performed using six different values of : mm/s ; mm/s [s]; mm/s (s); mm/s [s] and mm/s [s]. The evolution of the discharge process is assessed by repetitively moving the belt at the chosen constant velocity during a time interval s and by recording an image from the scanner while the belt is at rest.
For the present study the image analysis is used to determine the packing fraction, , and the number of grains, , that remain inside the confining frame at time . To do so, an intensity threshold is used to convert each image into binary: white is assigned to the rings (grains) and black is assigned to the background. Therefore, black disks at the center of each grains are isolated from one another, which makes it easy to detect them and to compute the number of grains remaining in the frame, , or, equivalently, the number of disks that flowed out the system at time , . The instantaneous flow-rate (averaged over s, because of the acquisition rate) is defined as .
The packing fraction is, by definition, the fraction of the surface area covered by the grains. In order to measure , the black disk at the center of the rings is filled with white in order to obtain white disks. The number of white pixels over the total number of pixels in the region of interest is a direct measurement of .
The reproducibility of the experiments has been checked by repeating the procedure up to three times for each set of the control parameters (, ).
Iii Experimental results
iii.1 Flow rate
The discharge process is analyzed as long as grains fill a distance of upstream of the outlet. We report the number of grains that flowed out the system, as a function of time . Two types of behavior are observed (Fig. 2):
For initially dense systems, increases linearly with the time . The flow-rate is constant
For initially loose systems, does not increase linearly with time . The flow-rate varies during the discharge.
The difference can be easily understood by considering that, for initially loose systems, the grains are progressively piling against the downstream wall Fig. 3. The discharge process can be thus described in two stages:
First stage (transient): the grains are piling progressively and the flow rate depends on time.
Second stage (steady): the system has reached a steady packing fraction, , slightly smaller than the maximum possible packing fraction (corresponding to the close packing), and the flow rate remains constant.
iii.2 Packing fraction
We expect the flow rate to be influenced by the packing fraction near the outlet. Therefore, we measure the packing fraction upstream of the aperture, in a region of width and thickness . The region under analysis is highlighted by a solid box in each of the images in Fig. 3.
During the discharge process, the grains pile progressively against the downstream wall until a steady state is reached. Accordingly, we observe that the packing fraction increases up to the asymptotic value, , slightly smaller than the value corresponding to the close packing (Figs. 4 and 5).
We observe that the temporal evolution of the packing fraction strongly depends on the initial packing fraction (Fig. 4) and, as expected, the asymptotic value is reached faster for larger belt velocities, . Indeed, for a given initial , all curves collapse when is reported against , the distance traveled by the belt at time (Fig. 5).
Iv Discussion and Conclusions
We aim here at accounting for the temporal evolution of the packing fraction in region close to the outlet, .
On the one hand, it is expected that the packing fraction increase, due to grains that enter the outlet region from the upstream region, at a rate which should be proportional to:
the belt velocity : the higher the value of the larger the income of grains from the upstream region;
the packing fraction in the vicinity upstream of the outlet, i.e the region enclosed in the dashed box in Fig. 3(b): a larger packing fraction indicates a larger amount of grains accessing from the upstream region;
the available space, thus to the difference between the and its maximum accessible value, : more available space allows a larger income of grains from the upstream region.
In addition, as can be observed in Fig. 3 (b), we can further assume that the packing fraction in the vicinity of the outlet does not differ significantly from that in the outlet region and we take . We thus write:
On the other hand, is expected to decrease, due to the grains that flow out through the aperture, at a rate proportional to:
: the higher the value of the larger the outflow from the system;
the local packing fraction : a larger packing fraction at the outlet indicates a larger amount of grains leaving the system.
where and .
Taking into account the initial condition that , the solution of Eq. (4) can be written in the form:
We point out that the prefactor is proportional to the belt velocity which provides the only timescale of the problem. This assertion is compatible with the observation of a nice collapse of the experimental data observed when the packing fraction in the outlet region is reported as a function of the distance traveled by the belt (Fig. 5). Therefore Eq. (5) can be rewritten as:
with a characteristic travel distance which is thus independent of the velocity. The measurements of (Fig. 5) are satisfactorily described by Eq. (6). For instance, the interpolation of the experimental data for all velocity leads to cm () and for . We indeed observe that the steady value of the packing fraction is smaller than as expected from our simple description of the problem.
Later, we will discuss the meaning of this characteristic length and its dependence with the initial packing fraction. But now, it is particularly interesting to analyze the potential effects of the changes in the local packing fraction on the flow rate. To do so, we consider that the flow rate is proportional to and .
We report in Fig. 6 the average velocity, , of the grains in the region upstream the outlet (Fig. 3). We display the average over the duration of the discharge. We observe that almost equals the belt velocity (to within the experimental uncertainty). No systematic dependence is observed as a function of , which indicates that this average is not altered by the presence or the absence of a transient. Therefore we can state that the characteristic velocity of the grains at the outlet remains approximately constant and equal to the belt velocity during the entire discharge. Moreover, we have observed that the instantaneous velocity, even if the measurements are noisier, does not significantly deviate from . Thus, the variations of the flow rate can only be attributed to the changes in the local packing fraction .
With the above statement in mind, we can replace the constant packing fraction in Eq. (1) by the time-dependent packing fraction given by Eq. (5). Doing so, we get the number of grains that left the system at time in the form:
with . A good agreement of experimental data with Eq. (7) (solid lines in Fig. 2) is observed. We found that ( cm) and, as will be explained below, we also observed that values of depend on . The agreement confirms that the typical velocity of the grains at the outlet to be considered in Berverloo’s law is not altered by the local packing fraction . It should also be noted that for initially dense systems the second term in Eq. (7) vanishes and Beverloo’s law (Eq.(1)) with a constant is retrieved: . Actually, in this case, and and a linear regression corroborates that cm.
As for the meaning of the characteristic length , it corresponds to the travel distance over which the system reaches the steady state (Eq. 6). It can be estimated by considering that the packing fraction, in a region above the downstream wall of typical height (which corresponds to the typical height of the arch that forms above the outlet), must have reached its steady-state value (of about ) for . In order to get a crude estimate, neglecting the outflow, one can assume that a region of height and packing fraction is compacted in a region of height and packing fraction , which leads to . This estimate is compatible with the increase of when is decreased (see Fig. 4) and with the absence of significant transitory for . In our experimental configuration, the outflow cannot be neglected as the width of the system is not much larger than the aperture size and the maximum packing fraction that can be reached is . In order to take into account the grains that escape the system, one can add a correction factor and write:
In summary, we have simultaneously measured the flow rate and the packing fraction in the outlet region of a discharging 2D-silo. We have observed that, for initially loose systems, the packing fraction in the outlet region evolves during the discharge and that, at the same time, the flow rate is not constant. We proposed that the flow rate is directly altered by the variations of the local density of the granular material and not by variations of the typical velocity at the outlet. This assertion is supported by a, simplistic, logistic model, accounting for the temporal evolution of both the packing fraction and the flow-rate, which proved to be in agreement with our experimental data.
Acknowledgements.This work has been supported by the program UBACyT (UBA) and the International Cooperation Program CONICET-CNRS. M. A. A. acknowledge support from CONICET.
Appendix: derivation of Eq. (7)
Therefore can be obtained by integrating the above expression between and :
The following substitution can be made with leading to:
which evaluated between and is:
Regarding that and :
So, we finally arrive to Eq. (7) by considering , i.e. there are no disks flowing out of the system at :
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