arXiv:physics/0610201v1 [physics.hist-ph] 23 Oct 2006 a emnpprfo 1852. from paper German nal theory hourglass ra .Tighe P. Brian -al [email protected] E-mail: USA, 27708, 90305, NC Box Durham, Physics, of Department University, Duke paper his in Hagen was ef it saturation 4], [3, the his silos for with granular credit in Janssen, fect receives While typically systems. paper, granular 1895 Ha- on is well-known work Less gen’s 2]. [1, law (Hagen-)Poiseuille well-known the now mea- is as his what studied pipes; 1839 in his in flow published for laminar surements on renowned study most the is to contributions Hagen Ludwig Heinrich Gotthilf Introduction 1 ndb iowls–gnrlykona the as con- known system generally granular – static walls satura a silo the in by is depth fined effect with first pressure The of matter: tion granular of fundamen- two aspects explained tal and measured Hagen Ludwig rich Abstract 2018 24, August Pape Hagen’s of Translation – Sand 1852 from Dry of Motion and Pressure evddrn h o u ftecnanr–tdyoften today – container ob- the dynamics of the out the describes flow called paper the his during of served part The spootoa otedaee fteoeigrie othe to raised opening the of discharg 5 diameter of power the rate the to that proportional discovers is and container, open- the an of through flow ing the detail considerable in examines later gen 40 than more Janssen exponen- by 7]. the [4, forward of put instead pressure form the tial for law cutoff) quadratic qualitative a some a proposes provided (with , Hagen that effect. first model the first the of a for understanding offered not and also – but [6] – cf. earlier Sandes trocknen effect des this Bewegung measured die und Druck den nefcieoeigdaee hti mle ytiethe twice by smaller finds is Hagen that [8]. diameter correlation opening Beverloo effective the an th resultin as of the known instead – is if used is best law diameter data effective the some fits diameter it real but 10.2]; ch [7, analysis wl eisre yteeditor) the by inserted be (will No. manuscript Matter Granular nadto otedsuso fasai ieo ad Ha- sand, of pile static a of discussion the to addition In / .Ti euti lgnl eie rmdimensional from derived elegantly is result This 2. narmral ae rm15,Gthl Hein- Gotthilf 1852, from paper remarkable a In eelolaw Beverloo · h olwn satasaino h origi- the of translation a is following The . atisSperl Matthias n om h onaino the of foundation the forms and – ase effect Janssen 5 who [5] Uber ¨ g e e - - . rsueaantteds equals disk the against pressure tn and stant ueo h iki atrmisucagdfrfilnshigher pres- fillings for the unchanged Hence, remains strong fact whole. in the a disk by the as on cylinder compensated sure the be on cannot exert acting part disk, friction lower bottom its the which throughout pressure, on connected axial rigidly the not therefore is and How- negative. cylinder even sand but zero the only ever, not become will after- it decrease wards; subsequently and maximum, a reach ginning, qaeo h egt Let is height. or the pressure, which it. horizontal of surrounding friction the square to sand the proportional the is less friction from it The cylinder above that weight sand the by by of experienced created cylinder disk the the on of exerted pressure a is there ftedaee fteopening. rescaling the the of by flow diameter the the of of successfu matter description a continuum a individual a provides is to Hagen the opening route [11], the of investigations at level current arching of the of probability on the While grains and [10]. un- fluid to flows devoted between granular been differences has fundamental work of the lot derstanding analy- a early recently, Hagen’s 10.4]. More extends [9]. sec and sis [7, confirms hourglass both an work al with Later thus time container, of the measure to in the found height lowing is filling material the granular of independent of be flow the where theory glass mea recent more with surements. consistent is which diameter, particle sa xeddfiln fsn pt height a it to of up top sand on of tightly; filling seals extended but an movable is easily is radius which disk includes of that opening bottom circular horizontal a a with container a Consider Sand Dry of Hagen, Ludwig Heinrich Gotthilf 2 o growing a For r ae’ okly u h ai ftes-aldhour- so-called the of basis the out lays work Hagen’s 2 πγ h − γ etewih faui oueo ad hnthe then sand, of volume unit a of weight the be 2 9hJnay1852 January 19th , r πγ lh h 2 . hsepeso ilices ntebe- the in increase will expression this , l eoeafito eedn con- dependent friction a denote r nti pnn spae a placed is opening this In . rsueadMotion and Pressure h saresult, a As . r the s - - , l 2 than that height at which the pressure on the disk reaches its [6]3. To reduce the mentioned irregularities as much as pos- maximum value. sible, I limited the of each observation to 30 to 200 For the case of pressures below the maximum, we seek to and, additionally, tried to make the fillings rather represent that pressure by the weight of a free-standing body uniform. To this end, the container was placed in a metal- of sand on the disk. The body is bounded by the surface of sheet cylinder with a sieve-like bottom, filled with sand, and the filling. It is a conoid that is formed by the rotation of a lifted slowly thereafter; in this way the sand poured into the parabola around its axis. Here the parameter of the parabola container in several hundred thin streams, each from a very is 4rl, while the height of the paraboloid is r/(4l). The latter small height. body joins the circumference of the opening. The openings at the bottom of the container, with their To compare these results with the real phenomenon, I sharp edges always on the upper face, had the following created openings of radii of 0.3791 inch1 and 0.7271 inch, radii: respectively, in two brass plates used as the the bottom of the opening 1 0.1677 inch sand-filled container; these openings were closed with suit- opening 2 0.1203 inch able disks that were supported from underneath by hooks opening 3 0.0986 inch that were connected to one arm of a balance, while the other opening 4 0.0807 inch arm carried the counterweight. To reduce the counterweight opening 5 0.0549 inch to the pressure on the disk slowly and without concussion, opening 6 0.0377 inch sand was allowed to discharge through a small hole in the The amount of outflowing sand per second, averaged bottom of the plate. The constant error of this method could over six observations each, was: be found easily by measurement of the excess weight of the for the opening 1 1.8995 Loth disk and the hook compared to the plate both by the dis- for the opening 2 0.7596 Loth charge of sand and by direct weighing. for the opening 3 0.4330 Loth for the opening 4 0.2481 Loth While repeating measurements of the pressure due to the for the opening 5 0.08242 Loth the sand against the disks multiple , considerable de- for the opening 6 0.02676 Loth viations among the measurements were observed; this was Comparing these weights with the radii of the openings, apparently due to different ways of settling. When the set- the former seem to be approximately proportional to the tling was as loose as possible, the weight of a cubic inch of third power of the latter. An attempt to represent them in the sand, a crude ferrous grit, was 2.9 Loth2. However, the the form weight increased to 3 Loth when there was modest agitation during filling and rose to 3.25 Loth as soon as a fairly com- m = kr3, pact settling was generated by severe agitation or by push- ing a wire into the container. In so doing, the friction in- however, did not produce a satisfactory result; the remaining creased by even more than the specific weight. Hence, the errors were rather significant and very regular, so they could pressure against the disk became remarkably smaller for the not be interpreted as observational errors. In contrast, if the more compact settling. radius of the opening was reduced by some length, a nearly With the larger disk the pressure maximum was reached constant ratio was observed between the mass of sand and at a filling height of around 1 inch: for larger height the pres- the 2.5th power of the reduced radius. The reduction of the sure decreased somewhat, since despite great care the sand radius is justified by noting that the granules that touch the settled in a slightly denser state. For the loosest fillings I edge of the opening while falling lose their speed partially or found l = 0.154 to 0.175. In contrast, the deviations were completely and even disturb the motion of the neighboring less when the sand was dropped from a height of several granules when bouncing off. From repeated measurements it inches in a narrow stream, and when the sand flow was cone- was found that, on average, 9 granules of sand constitute the shaped: the value for l was limited to between 0.21 and 0.22. length of a Rhineland line4 hence the diameter of a single The influence of different settling states was also clearly one equals 0.0093 inches. noticeable as the sand discharged through the openings of Thus I compared the masses of the sand with the expres- various radii. When the settled filling was somewhat denser, sion there was less sand flowing within a second. As the dis- 5 − 2 charge duration increased, the sand became agitated, par- m = k(r x) ticularly in the vicinity of the opening, resulting in an in- and found after introducing approximate values for x accord- creased sensitivity, which in turn led to a diminished flow. ing to the method of least squares from all six observations Incidentally, the height of the filling had no influence, as was already recognized by Huber-Burnand some time ago k = 189.07 x = 0.00968. 1 trans. note: The German Zoll is translated as inch; however, it might deviate slightly from the value generally used today; most likely 3 trans. note: The reference was added for the translation. There are 1 Zoll = 26.15mm [12]. no references given in the original. 2 trans. note: The German Loth (or Lot) is a unit of mass, 1 Loth = 4 trans. note: The German Rheinl¨andische Linie is a unit of length, 1/32 Handelspfund(Berlin) ≈ 14.6g [12]. typically 1/12 of a Zoll, hence 1 line = 2.18mm [12]. 3

Using these values to calculate m, the remaining errors are If instead one introduces the value for 1 -0.0181 for 2 +0.0094 l = 0.225, for 3 +0.0135 for 4 +0.0075 which is valid for packings where sand is flowing sideways, for 5 -0.00003 which really happens during the discharge, then for 6 -0.00142 h = 0.4938 · r. The first observation agrees the least with the fitted val- ues, but is also in itself less accurate than the other obser- The result derived from the observations is in between the vations for two reasons: for one part, its duration was the two when r is interchanged with ρ. This interchange is nec- shortest, and for the other part, the sand was poured out essary because the sand only hits the edge of the opening vehemently and became heavily agitated, and the container when in motion; in contrast, when at rest all the sand grains was almost completely emptied. Therefore, I tried to deter- encountering the movable disk also load it. This confirms the mine the values of the two constants independently of the assumption from above that the free fall of the sand starts on first observation, using only the last five. This gives the surface of the paraboloid; it also explains that the amount

5 of sand flowing through the opening is proportional to the m = 181.57(r − 0.00893) 2 . 5 power 2 of the effective radius of the opening. Finally, some comments should be made about the mo- The deviations from the observed weights were thereupon tion of the sand during discharge. for 1 -0.0712 In the four inch wide and ten inch tall container, above for 2 -0.0085 the outlet, the entire surface of the sand packing subsided for 3 +0.0050 uniformly in the beginning. Only gradually did a dip form for 4 +0.0039 vertically above the outlet. The dip grew continually, and for 5 +0.00002 above its sides sand fell down. Concurrently, at the rim of the for 6 -0.00 078 container a ring-shaped, almost horizontal surface remained. The radius reduction x for the discharge opening is found This surface also subsided, but without the granules of sand to be close to the diameter of a grain of sand. From the experiencing strong sideways motion. The flat ring gradually constant k can be determined the average distance from the assumed a smaller width and disappeared completely when opening at which the sand begins its free fall. Assuming that the funnel-shaped dip reached the outlet. From this it follows the sand forms a compact mass until reaching the opening, that the sand flows not only vertically towards the outlet but the amount of sand discharged per second is also along concentric inclined trajectories, and that the mo- m = 2ρ2πγpgh tion extends up to a slope, which a free surface of sand can exhibit. where h designates the mentioned height of fall and ρ the The motion in the inner part of the sand mass revealed effective opening. On the other hand, the observations yield itself very explicitly when I filled sand in a container hav- ing side walls made from a glass panel. Since this glass 5 m = 181.57 · ρ 2 . panel touched the outlet, one could follow the motion of sin- gle grains of sand down to the opening. The strongest flow Equating both expressions and setting γ to 2.93 as obtained formed vertically above the outlet; in fact the sand granules from the average loose packings, one finds approached it with increasing yet moderate speed until, di- rectly above, they were accelerated in a way that they could h 0 5185 · ρ = . . no longer be seen. Nevertheless, the sand also flowed in- If, in contrast, one assumes that for each wards from the side of the outlet, but this motion was inter- a layer of sand of the same vertical height separates from rupted frequently and only occurred periodically, presum- the whole inner area of the paraboloid mentioned above in ably due to friction at the glass. a free fall, then one can easily find the average velocity of Underneath the opening, the stream of sand was not nearly this layer while passing the opening, and from the latter the as sharply bounded as a water-jet; rather, it was surrounded average height of fall of the entire mass. This height is by single granules that from time to time departed as far as several lines5. Because of this the streams flowing from the r h = . larger outlets showed a significant reduction in their diam- 9l eters, extending about 2 inches deep. In addition, the mea- But from the data for loose packings it was found that surement showed that even immediately under the disk the stream is already much weaker than at the outlet. The out- l = 0.16 let was 0.335 inch in diameter while the stream was only 1 0.29 inch at a distance of 1 2 lines, and contracted to 0.27 inch and hence at greater depth. The reason for this effect is not the effective h = 0.6944 · r. 5 trans. note: Linien, plural as in Rheinl¨andische Linien. 4 reduction of the opening mentioned above, since this would only explain the weakening of the stream by about 2% of an inch; instead, the sand flowing sideways continues its mo- tion towards the axis even after having passed the outlet, and the granules hitting the rim of the opening are also reflected towards the axis. The stream of sand hence experiences a contraction sim- ilar to the stream of a liquid; and when one compares the diameter of the opening with the smallest diameter of the stream, the ratio appears as 1 : 0.806 or for the ratio of cross sections 1 : 0.650. This agrees closely with the known contraction ratios for liquid streams leaving openings in thin walls.

Acknowledgements This work is supported by nsf-dmt0137119, DMS0244492, and DFG SP 714/3-1.

References

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