arXiv:physics/0610201v1 [physics.hist-ph] 23 Oct 2006Granular Matter manuscript No.(will be inserted by the editor)Brian P. Tighe Matthias SperlPressure and Motion of Dry Sand Translation of Hagens Paperfrom 1852February 2, 2008Abstract In a remarkable paper from 1852, Gotthilf Hein-rich Ludwig Hagen measured and explained two fundamen-tal aspects of granular matter: The rst effect is the satura-tion of pressure with depth in a static granular system con-ned by silo walls generally known as the Janssen effect.Thesecondpart ofhispaperdescribesthedynamicsob-served duringthe ow outof the container today oftencalled the Beverloo law and forms the foundation of thehourglass theory. The following is a translation of the origi-nal German paper from 1852.1 IntroductionGotthilf Heinrich Ludwig Hagen is most renowned for hiscontributions to the study on laminar ow in pipes; his mea-surements published in 1839 studied what is nowwell-knownas the (Hagen-)Poiseuille law [1, 2]. Less well-known is Ha-gensworkongranularsystems. WhileJanssen, withhis1895 paper, typically receives credit for the saturation ef-fect in granular silos [3, 4], it was Hagen in his paperUberden Druck und die Bewegung des trocknen Sandes [5] whomeasured this effect earlier but also not for the rst time,cf. [6] and offered a rst model that provided a qualitativeunderstanding of the effect. Hagen proposes a quadratic law(with some cutoff) for the pressure instead of the exponen-tial form put forward by Janssen more than 40 years later[4, 7].In addition to the discussion of a static pile of sand, Ha-gen examines in considerable detail the owthrough an open-ing of the container, and discovers that the rate of dischargeis proportional to the diameter of the opening raised to thepower 5/2. This result is elegantly derived from dimensionalanalysis [7, ch 10.2]; but it ts the data best if instead of thereal diameter some effective diameter is used the resultinglaw is known as the Beverloo correlation [8]. Hagen ndsan effective opening diameter that is smaller by twice theDuke University, Department of Physics, Box 90305,Durham, NC 27708, USA,E-mail: msperl@duke.eduparticle diameter, which is consistent with more recent mea-surements.Hagens work lays out the basis of the so-called hour-glass theory where the ow of granular material is found tobe independent of the lling height in the container, thus al-lowing the measure of time with an hourglass [7, sec 10.4].Later work both conrms and extends Hagens early analy-sis [9]. More recently, a lot of work has been devoted to un-derstanding the fundamental differences between uid andgranularows[10]. Whileonthelevel oftheindividualgrains the probability of arching at the opening is a matterof current investigations [11], Hagen provides a successfulroute to a continuum description of the ow by the rescalingof the diameter of the opening.2 Gotthilf Heinrich Ludwig Hagen, Pressure and Motionof Dry Sand, 19th January 1852Consider a container with a horizontal bottom that includesa circular opening of radius r. In this opening is placed adisk which is easily movable but seals tightly; on top of itis an extended lling of sand up to a height h. As a result,there is a pressure exerted on the disk created by the weightof the cylinder of sand above it less the friction which isexperienced by that cylinder from the sand surrounding it.The friction is proportional to the horizontal pressure, or thesquare of the height. Let l denote a friction dependent con-stant and be the weight of a unit volume of sand, then thepressure against the disk equalsr2h2rlh2.For a growing h, this expression will increase in the be-ginning, reach a maximum, and subsequently decrease after-wards; it will become not only zero but even negative. How-ever, the sand cylinder is not rigidly connected throughout,and therefore the axial pressure, which its lower part exertson the bottom disk, cannot be compensated by the strongfriction acting on the cylinder as a whole. Hence, the pres-sure on the disk in fact remains unchanged for llings higher2than that height at which the pressure on the disk reaches itsmaximum value.For the case of pressures belowthe maximum, we seek torepresent that pressure by the weight of a free-standing bodyof sand on the disk. The body is bounded by the surface ofthe lling. It is a conoid that is formed by the rotation of aparabola around its axis. Here the parameter of the parabolais 4rl, while the height of the paraboloid is r/(4l). The latterbody joins the circumference of the opening.To compare these results with the real phenomenon, Icreated openings of radii of 0.3791 inch1and 0.7271 inch,respectively, in two brass plates used as the the bottom of thesand-lled container; these openings were closed with suit-able disks that were supported from underneath by hooksthat were connected to one arm of a balance, while the otherarm carried the counterweight. To reduce the counterweightto the pressure on the disk slowly and without concussion,sand was allowed to discharge through a small hole in thebottom of the plate. The constant error of this method couldbe found easily by measurement of the excess weight of thedisk and the hook compared to the plate both by the dis-charge of sand and by direct weighing.While repeating measurements of the pressure due to thethe sand against the disks multiple times, considerable de-viations among the measurements were observed; this wasapparently due to different ways of settling. When the set-tling was as loose as possible, the weight of a cubic inch ofthe sand, a crude ferrous grit, was 2.9 Loth2. However, theweight increased to 3 Loth when there was modest agitationduring lling and rose to 3.25 Loth as soon as a fairly com-pact settling was generated by severe agitation or by push-ing a wire into the container. In so doing, the friction in-creased by even more than the specic weight. Hence, thepressure against the disk became remarkably smaller for themore compact settling.With the larger disk the pressure maximum was reachedat a lling height of around 1 inch: for larger height the pres-sure decreased somewhat, since despite great care the sandsettled in a slightly denser state. For the loosest llings Ifoundl= 0.154 to 0.175. In contrast, the deviations wereless when the sand was dropped froma height of severalinches in a narrow stream, and when the sand ow was cone-shaped: the value for l was limited to between 0.21 and 0.22.The inuence of different settling states was also clearlynoticeable as the sand discharged through the openings ofvarious radii. When the settled lling was somewhat denser,therewaslesssandowingwithinasecond. Asthedis-chargedurationincreased, thesand became agitated, par-ticularly in the vicinity of the opening, resulting in an in-creased sensitivity, which in turn led to a diminished ow.Incidentally, theheightof thelling had noinuence, aswas already recognized by Huber-Burnand some time ago1trans. note: TheGermanZoll istranslatedasinch; however, itmight deviate slightly from the value generally used today; most likely1 Zoll = 26.15mm [12].2trans. note: The German Loth (or Lot) is a unit of mass, 1 Loth =1/32 Handelspfund(Berlin) 14.6g [12].[6]3. To reduce the mentioned irregularities as much as pos-sible, I limited the duration of each observation to 30 to 200seconds and, additionally, tried to make the llings ratheruniform. To this end, the container was placed in a metal-sheet cylinder with a sieve-like bottom, lled with sand, andlifted slowly thereafter; in this way the sand poured into thecontainer in several hundred thin streams, each from a verysmall height.The openings at the bottom of the container, with theirsharpedgesalwaysontheupperface, hadthefollowingradii:opening 1 0.1677 inchopening 2 0.1203 inchopening 3 0.0986 inchopening 4 0.0807 inchopening 5 0.0549 inchopening 6 0.0377 inchTheamount ofoutowingsandpersecond, averagedover six observations each, was:for the opening 1 1.8995 Lothfor the opening 2 0.7596 Lothfor the opening 3 0.4330 Lothfor the opening 4 0.2481 Lothfor the opening 5 0.08242 Lothfor the opening 6 0.02676 LothComparing these weights with the radii of the openings,theformerseemtobeapproximatelyproportional tothethird power of the latter. An attempt to represent them inthe formm = kr3,however, did not produce a satisfactory result; the remainingerrors were rather signicant and very regular, so they couldnot be interpreted as observational errors. In contrast, if theradius of the opening was reduced by some length, a nearlyconstant ratio was observed between the mass of sand andthe 2.5thpower of the reduced radius. The reduction of theradius is justied by noting that the granules that touch theedge of the opening while falling lose their speed partially orcompletely and even disturb the motion of the neighboringgranules when bouncing off. From repeated measurements itwas found that, on average, 9 granules of sand constitute thelength of a Rhineland line4hence the diameter of a singleone equals 0.0093 inches.Thus I compared the masses of the sand with the expres-sionm = k(r x)52and found after introducing approximate values for x accord-ing to the method of least squares from all six observationsk = 189.07x = 0.00968.3trans. note: The reference was added for the translation. There areno references given in the original.4trans. note: The German Rheinl andische Linie is a unit of length,typically 1/12 of a Zoll, hence 1 line = 2.18mm [12].3Using these values to calculate m, the remaining errors arefor 1 -0.0181for 2 +0.0094for 3 +0.0135for 4 +0.0075for 5 -0.00003for 6 -0.00142The rst observation agrees the least with the tted val-ues, but is also in itself less accurate than the other obser-vations for two reasons: for one part, its duration was theshortest, andfortheotherpart, thesandwaspouredoutvehemently and became heavily agitated, and the containerwas almost completely emptied. Therefore, I tried to deter-mine the values of the two constants independently of therst observation, using only the last ve. This givesm = 181.57(r 0.00893)52.The deviations from the observed weights were thereuponfor 1 -0.0712for 2 -0.0085for 3 +0.0050for 4 +0.0039for 5 +0.00002for 6 -0.00 078The radius reduction x for the discharge opening is foundtobeclosetothediameter ofagrainofsand. Fromtheconstant k can be determined the average distance from theopening at which the sand begins its free fall. Assuming thatthe sand forms a compact mass until reaching the opening,the amount of sand discharged per second ism = 22

ghwhere h designates the mentioned height of fall and theeffective opening. On the other hand, the observations yieldm = 181.57 52.Equating both expressions and setting to 2.93 as obtainedfrom the average loose packings, one ndsh = 0.5185 .If, in contrast, oneassumes that for each unit of timea layer of sand of the same vertical height separates fromthe whole inner area of the paraboloid mentioned above ina free fall, then one can easily nd the average velocity ofthis layer while passing the opening, and from the latter theaverage height of fall of the entire mass. This height ish =r9l.But from the data for loose packings it was found thatl = 0.16and henceh = 0.6944 r.If instead one introduces the valuel = 0.225,which is valid for packings where sand is owing sideways,which really happens during the discharge, thenh = 0.4938 r.The result derived from the observations is in between thetwo when r is interchanged with . This interchange is nec-essary because the sand only hits the edge of the openingwhen in motion; in contrast, when at rest all the sand grainsencountering the movable disk also load it. This conrms theassumption from above that the free fall of the sand starts onthe surface of the paraboloid; it also explains that the amountof sand owing through the opening is proportional to thepower52 of the effective radius of the opening.Finally, some comments should be made about the mo-tion of the sand during discharge.In the four inch wide and ten inch tall container, abovethe outlet, the entire surface of the sand packing subsideduniformly in the beginning. Only gradually did a dip formvertically above the outlet. The dip grew continually, andabove its sides sand fell down. Concurrently, at the rimof thecontainer a ring-shaped, almost horizontal surface remained.This surface also subsided, but without the granules of sandexperiencing strong sideways motion. The at ring graduallyassumed a smaller width and disappeared completely whenthe funnel-shaped dip reached the outlet. From this it followsthat the sand ows not only vertically towards the outlet butalso along concentric inclined trajectories, and that the mo-tion extends up to a slope, which a free surface of sand canexhibit.The motion in the inner part of the sand mass revealeditself very explicitly when I lled sand in a container hav-ingsidewallsmadefromaglasspanel. Sincethisglasspanel touched the outlet, one could follow the motion of sin-gle grains of sand down to the opening. The strongest owformed vertically above the outlet; in fact the sand granulesapproached it with increasing yet moderate speed until, di-rectly above, they were accelerated in a way that they couldno longer beseen. Nevertheless, the sand also owed in-wards from the side of the outlet, but this motion was inter-rupted frequently and only occurred periodically, presum-ably due to friction at the glass.Underneath the opening, the streamof sand was not nearlyas sharply bounded as a water-jet; rather, it was surroundedby single granules that from time to time departed as far asseveral lines5. Because of this the streams owing from thelarger outlets showed a signicant reduction in their diam-eters, extending about 2 inches deep. In addition, the mea-surement showed that even immediately under the disk thestream is already much weaker than at the outlet. The out-let was 0.335 inch in diameter while the stream was only0.29 inch at a distance of 112 lines, and contracted to 0.27 inchat greater depth. The reason for this effect is not the effective5trans. note: Linien, plural as in Rheinl andische Linien.4reduction of the opening mentioned above, since this wouldonly explain the weakening of the stream by about 2% of aninch; instead, the sand owing sideways continues its mo-tion towards the axis even after having passed the outlet, andthe granules hitting the rim of the opening are also reectedtowards the axis.The stream of sand hence experiences a contraction sim-ilar to the stream of a liquid; and when one compares thediameter of the opening with the smallest diameter of thestream, the ratio appears as1 : 0.806or for the ratio of cross sections1 : 0.650.This agreesclosely with the knowncontraction ratiosforliquid streams leaving openings in thin walls.Acknowledgements This work is supported by nsf-dmt0137119, DMS0244492,and DFG SP 714/3-1.References1. G. H. L. Hagen, Poggendorffs Annalen der Physik undChemie 46, 423 (1839).2. S. P. Sutera and R. Skalak, Annu. Rev. Fluid Mech. 25,1 (1993).3. H. A. Janssen, Zeitschr. d. Vereinesdeutscher Inge-nieure 39, 1045 (1895).4. M. Sperl, Granular Matter 8, 59 (2006).5. G. H. L. Hagen, Bericht uber die zur Bekanntmachunggeeigneten Verhandlungen der K oniglich PreussischenAkademie der WissenschaftenzuBerlin. pp. 3542(1852).6. Huber-Burnand, Ann. d. Phys. 92, 316 (1829).7. R. M. Nedderman, Statics and kinematics of granularmaterials (Cambr. Univ. Press, Cambridge, 1992).8. W. A. Beverloo, H. A. Leniger, andJ. van deVelde,Chem. Eng. Sci. 15, 260 (1961).9. K. Wieghardt, Annu. Rev. Fluid Mech. 7, 89 (1975).10. J. Rajchenbach, Adv. Phys. 49, 229 (2000).11. K. To, P.-Y. Lai, and H. K. Pak, Phys. Rev. Lett. 86, 71(2001).12. W. Trapp, Kleines Handbuch der Mae, Zahlen,Gewichte und der Zeitrechnung (Reclam, Stuttgart,1996).