An experimental device for measuring the time averaged momentum associated with a steady stream of a disperse granular material has been developed. The mathematical basis for the device is presented including a discussion of using the momentum measurement to compute the local mass or energy fluxes. The analysis considers both nonuniform particle mass and nonuniform velocities for the various constituents of an aggregate material. The results of calibration experiments conducted with a prototype transducer are shown with theoretical predictions of these results

Griffiths, Stewart K.

Watling, Hugh E.

UNT Digital Library

.,. • UCRL- 87243 PREPRINT Ci- s;;,0514' --3 An Experimental Device for Measurinq The Momentum of Disperse Granular Materials Hugh E. !fotling Mechanical Enqineerin0 Depart~ent Lawrence Livermore National Laboratory Stewart K. Griffiths Sandia National Laboratories A 1 buquerque, New ~1ex i co 1982 Joint Conference on Experimental Mechanics Honolulu, Hawaii May 23 - 30, 1982 February 10, 1982 This is a preprint of a paper intended for publication in a journal or proceedings. Since changes may be made before publication , this preprint is made available with the un-derstanding that it will not be cited or reproduced without the permission of the author . DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. ,. , UCRL--87243 ABSTRACT I. - . DISCL:M:i - . --- -- ~ This book was prepared as an account ol work sponsored by an agency of the Uni1ed States Government. Neither 1he United States Government nor any agency th~r~f,' nor any of th~i~ .employees, makes any :;~~et~~=~r:S ~~,~~:~ed~ta::Ui':r:i~o~~~P~!~~t:. ~~~=n~~,h~~ 1:~~:[.a~; ~ represents that its use v.<1uld not infringe privately owned rights. Reference herein to any specific commercial product, process. or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agencv thereof. The views and opinions of authors ei1pressed herein do not necessarily state or reflect those ol the United States Government or any agency thereof. AN EXPERIMENTAL DEVICE FOR MEAStif<lliG THE MOMENlU~' or OJSP[RS[ GRAl•UL;.R l·:.:.iERlALS • Hugh [. \.lat ling Mechanical [naineerina Department Lawrence Livermo~e National Laboratory Livermore, California Stewart K. Griffiths Sandia National Labpratorie~ Albuquerque, New Mexico · An experi~ental device for measuring the time averaged momentum a~sociated with a steady stream of a disperse granular material has been developed. This paper presents the mathematical basis for the device and includes a discussion of using the momentum measurement to compute the local mass or energy fluxes. The analysis considers.bo~h nonuniform particle m~ss a~a nonuni~orm velocities fo~ the various constituents of an aggregate materlal. The results of callbratlon t·~perlments conducted wlth a prototype transducer are shown with theoretical predictions of these results. I NTROOUCT JON Consider a quasi-.steady, disperse stream of particles intersecting a plane such that a finite amount of mass crosses the plane in some time interval. The particles comprising this mass may be nonuniform in size and in velocity. A significant physical Quant it~ which could be used to characterize such a stream is the time averaged momentum flux. [Qulvalently, this problem can be_posed by considering a collection of particles approaching and colliding with a solid surface, rather than crossing an imagined plane. The corresponding momentum flux is the flux incident of the surface provided that the particles being reflected by the surface do not interact with these of the incoming stream. The local, time averaged flux incident on a surface can, therefore, yield a characterization of the undisturbed stream. lf the mass and velocity distribution of the particles in a strrAm are known, the momentum flux described above may be used to compute the time-averaged mass flux crossing the plane or incident'·on the surface. The mass flux, like the momentum flux, is .necessarily time averaged. The mass flux of a sin!JlP triln~l11ting particle is at best very poorl.Y defi.ned. One particle is either cut by the plane or it is not. As a consequence, the instantaneous mass flux of a num~er of particles sequentially. crossing a plane appears as a series of infinite (or very large) pulses separated by zero values. Jn many cases, the instantaneous mass flux is not of primary i"nterest. A ml're useful quantity may be the time averaged flux. This average is obtained by integrating the instantaneous flux o~er some time int~rval and dividing the result by the magnitude of the interval. That the momentum flux incident on a surface is the flux of the undisturbed strea~ permits characterization of the undisturbed stream by measuring the i11'4lactfon force of the particles striking the surface. A device based on this principle has been developed and tested. The following sections outline major aspects of the mathematical basis of the device. The results of calibration experiments are presented with the theoretical results. GOVERNING EQUATIONS The classic theory of impact is based on the impulse-momentum l.,w for riaid bl'dies. Governing relations are well known [l). For linrar m(lmentum, in vector nota,tion the!.e.are ti(mV) • m(v1 - v0 ) ~/t1 f dt to •1h1s work ..-as performed under the ausplCcS of the U. 5. Department of Energy by Lawrence Livermore Laboratory under contract No. W-7405-Eng-48. ( l) ( 2 .... ·. M is the mass, and V is the linear velocity, f. is the force acting on the body, and t is time. The subscripts O and 1 indicate.initial and final conditions. The times to and t1 indicate the beginning and end o( the time interval over which the force is applied. In one dimension, Eq. (1) becomes .... .... t .... m(Vlz - VOz) =~ 1 fz dt to. (2) We note that the impulse-momentum relations do not require nor imply conservation of energy. They are Valid independent of whether or not .the impact is elastic. Furthermore, we note that the equations do not involve material properties as such. Although the contact time, ( 3 )' will depend on material properties, these properties we note that the times to and ti need not be known. particle is clearly zero when the particle is not in integration in the impulse-momentum equations may be Eq. 2 could be written do not enter the equations explicitly, Finallyi Because the external force acting on.the' contact with the surface, the limits 6f .. extended to ~lus or minus infinity. For example,· (4) The result in either case is the same. The simple relations presented above can be extended into a more useful domain by considering a particle impacting on a surface that is part of a classic spring-mass-damper system. A diagram of such a system is shown in Fig. l. The problem ·now becomes one comprised of an impusle coupled to a system response or "ringing". When the body mass Mb is much greater than the particle.mass m, it is reasonable to assume that t_.he force exerted on the body by the particle is that which would be exerted on a rigidly held surface fz. Therefore, the equation governing the system response is · F - c z - Mb z = 0 (5) where a dot denotes differentiation with respect to time and . F =-kz (6) subject to some bounded initial conditions at time t1. For any such initial conditions; the solution of Eq. 5 exhibits the properties [2]. z .... 0 as t .... co (7) and z .... 0 as t .... co (8) Integrating Eq. 5 once with respect to time, taking the limit as t goes to infinity and making use of Eq. 7 and Eq. 8 gives t . ·. t ~ limj Fdt=limj fzdt t....,., t0 · t....,., t0 It is now possible to express the net change in linear momentl,lm of a si.ngle particle in terms of the force measured in a spring-mass-damper system. (9) 3 ~ t ~ t ti(mV ) = ( l f dt = f 2 F dt z lt z t n 0 ( 10) . provided that ti and t2 are sufficiently large such that z and z are small. Now consider an isolated multi-pulse series resulting from a finite number of particles impacting on a surface. The impact force due to any particle will·add linearly with that of any other particles ~ ~imultaneously in contact with the surface. Therefore, the net change in linear momentum of all i ~articles of a multiple particle series is simply the integral of the impact force over the total tim~ interval of the series. The resulting impulse-momentum relation for an isolated multiple impact series \ is + . t + t ~ ti(m. V . ) = j 1 f dt = ( l F dt l l Zl t Z jt 0 0 ( 11) Dividing each sid~ of Eq. 11 by the time interval tit01 , the two integral terms are recognized as the time averages of fz and F. Now, let Nj denote the number of particles whose size [3] lies in the range daj to daj+l which impact in the time interval tJ.to1~ Assuming that 6Vzi and mi are smooth, continuous functions of dai• Eq. 11 can be surmned over j as + l '" N m Av .· = F 6t01 J j j u ZJ where a bar denotes time averaging and where mj and 6.Vzj are evaluated at daj· The total mass impacting in the time interval 6.to1 is simply and the mass fraction of particles in the range daj to daj+l which impact in the time interval is SJ. : N. m ./i: N. in. J J j J J -;-With these two definitions, Eq. 12 can be written in terms of the time averaged mass flow rate M. = F (12) ( 13) (14) (15) For a steady, fully-developed disperse stream of particles the time averaged mass frar.tion of particles in a given S'iZe range passing a plane or impacting on a surface is the same as the bulk mass fraction in that range (i.e., as determined by sieve. analysis). That is, 8. = n . = <Pm ( d ·+l ) - <Pm ( daj) J J aJ where nj is the bulk mass fraction of the particles which lie in the size range daj to daj+l ~nd dm i5 the cumulative distribution by mass. With Eq. 16, the form of Eq. 15 suggests an integral. Taking the limit as daj+l -daj approaches. zero gives, The frequency distribution by mass fm is. by definition, 'fm (da): lim n.//J.d. tid .+O J aJ aJ Th1s function is well-defined for most particle systems and for any usual granular material. ( 16) ( 17) (18) lhe intf•gr<il in EQ. 17 is, for steady-state co11dit ions, a const~nt. Thus, the time avera\H'd mass flow rat!' has been related to the time averagrct impact force on the pclrticles or the force as mrasureo in a spring-mass-damper system. If the out-going velocity of C8Ch particle is nearly zero (either slightly positive.or negative}. then the rehtion depends only on their incoming velocities. Based on the force measured in a spring-m11ss-di1mper systE'm, we then have 4 i1 = F ('' fm (d ) v0 (d ) dd 10 a z a a (19) For particles in free-fall, the vertical velocity, as a function of size, can i-e'adily be co!'IPule~. The rioht side of EQ. 19, except F, can therefC'lre be known in terms of properties of the particle'systerii"~r oranular material only. We note that the integral term in EQ. lg is the eQuivalent momentum velpt'ji'y": [.:j definE'd as (20) Thus, the time averageo momentum transport rate is (21) Similarly, the eQuivalent energy velocity and corresponding energy transport rate.can be defined. ( 22) THE DEVICE E (23) A orawino of the instrument is shown in Fig. 2. It consists of two cylinders, one inside the other. The inner cylinder 'is supported on a hydraulic piston/cylinder attached to the outer cylinder. A pressure transducer is located at the bottom of the fluid reservoir. By measuring the fluid pressure, the vertical force applied to the inner cylinder carr be detrrmined. f = PfAP Pt_ is the fluid pressure and Ap is the arra of the piston. For the prC'totype, Ap;, .11 in2 [.71 cm'). The electric o~tput of the transoucer is related to the pre~sure by its sensitivity. t • S Pr where S = 2.0 mV/PSJ [.29 ~V/KPAl for thE' prntntypc. ( 211) (25) The inner cylinder contains t'wo baffles; a conic section along the cc>nter line of the cylinder ·and a beveled ring along its inner prrin1Pter. lhcse are arranged so tt1Jt any· particle entering tile top of·· the cylinder will strH.c> one or hoth of the baffles and then e:dt thq1u9h the bottom with a negligible velccity. Thus, the change in velocity of any particle, as it pas~es through the device,· is eQual to its initial incoming velocity. All of its momc>ntum·is transferr('d to the inner cylinder. l.'ith these definitions ilnd prrvious results, the theoretical <'kctric output can be computed from eQuations 21, t4, and 25. c • or M = S ~ \;-p/Ap (rl\p)/(SVp) ( 76) (27) 5 (quat ion 27 <1('sc1·it1es the toul time· avercl9Ps mclSS flow through thr orvice and not the mass flux. The flu"' is ol•taineo by dividng each side by the cross-section area of the inner cylincier Ac. Ac {5 in2 [ 160 cm2] for the prototype. When the mass f lo" r~te or flux arr ~nown, the corresponoing n;cmentum or energy can bl- computed from equations 21 or 23. CALJSRATJON AND TESTS The prototype device ~-as tested using LLNL coarse fill (pea Q·•·avel) [3,5). The- material was dispensed from a container, through an orifice, 23 ft. [7.0 M) abovr thr device. For free-fall from this elevation, the eouivalent mClmentum velocity for LLNL coarse fill h [4) \'p s 34 ft/s (JOm/s~ The time-aver~9eo mass flow rate was determined from the total time of a test, the volu~e of material used anc its bulk densi~y. Using various size orifices, mass flow rates from abou} .2t -~~ ~.O lbm/s [. 11 to l.~ k9/S) 1>ere achieved. · · · . .- · Three methods of reducing the rough data to obtain.a single timc-~vrraged Vcllue of the ~lectric output have been used. These were digital time avera9ing; high frequency analo9 fi lterin9 and simple curve fit through the strip chart record. All three 9ave nearly identical results in several test cises. The third methco was adopted for general use. The results of calibration tests are shown in Fio. 3. The theoretical response is also shown. The aoreement seems very 0000 over the range of flow rates .. considered. At a mass flow rate of about ~:c lbm/sec [l.S k9/s) the instrument begins to accumulate mateiial which eventually fills the inner cylinder. During this transition, the cievice is wei9hing the accumulated gravel and measurement of the iffipact force is no longer accurate. The approximate rate at which this occurs is noted on the plot as choking. CONCLUSIONS An experimental device for measuring the momentum associated with a steady stream of a disperse oranular material has been developed and tested. The measurement madr with this device can be used to compute the mass, momentum or energy flux of such a stream. T.ests have sho~·n that measurements made when the mass flow rate is known are in good agreement with theoretical predictions of the.response. REFERENCES 1. Golosmith, W., Jrnpact, Edward Arnold (Publisher) Ltd., London, 1%0. 2. Ritger, P.O. ano Rose, N.J., Differential Eouations with ApplicatiClns, McGraw, Hill Book Co., New York, 1960. 3. Griffiths, S. K., The Size Distribution of LLNL Coarse Fill, UCID-1~~86, Lawrence Livermore National Laboratory, Livermore, California 1981. 4. Griffiths, S. I\._, The Velocity of LLNL Coars_~_l_i_l] in Free Fall, intrrnel report, Lawrence Livermore National Laboratory. 5. l>atling, H. E., _Spl-cification: LLNL Coar~s Fil], Lawrence Livrrmrrr N11tinn11l I 11hnr<1tory, Livermore, California, MEL-78-001400, 1S78. .. ~-' . , -~.; ~ ----··•· ----- -0 Fiqure l F= - kz ... v 1z PISTON ASSEMBLY TRANSDUCER -· 50 PSI 100.0 40.0 ..... fl) ·-'"'" ..J 0 20.0 > E 10.0 ·-TUBE , STEEL 6 IN. 0.0. x 3 I 16 ·IN. WALL 1.r.----TUBE • STEltL CHOKING 6 3 I 4 IN. O.D. x 1 I 4 IN. WALL F_igure 2 THEORETICAL RESPONSE I-::::> Cl. I-:::i 0 2.0 o., 0.2 0.4. 1.0 2.0 4.0. 10.0 MASS FLOW RATE - M ( lbm/eec ) ( THROUGH DEVICE ) Fiqure 3 "· ' l>ISCI. \1\11-.H Thi' dornml'nl '"" pri·pari·d "' an arrounl of "orl. 'Jl"n"1ri·d h_1 an :tl,!l'nc_1 nf lht· l 'ni11·d S1a11·' Cmnnnll'nl. '\l'iilll'r 1h1• l nir,•d S1a11·' (;,11,·rnm<·n1nor1ht· l niHf'il_1 of California nor an_1 of lht·ir •·111pl11_1l'l''· mal-1·' an' "arranl\. l'\-P"'" or implil·d. or a"um1·' an) lq:al liahilil' or " ' 'Jl""'ihili1_1 for llH' ac-rnrnc_1. rompll'l1·n1·''· or u'duln"" of an_1 i11formali1111, apparalu,, product. or prn1·1·" di,clo"·d, or r<·pn·"·ni- that ii- U"' \\ould nol infrinJ.!l' pri,a1t·l_1 O\\nt·d ril,!hl,. Ht•f1•n ·nct• h<•r<· in to an) 'IH'rifir l'omnH·rcial products. pron·"· or'''" ict• In lratk nanH'. lradl'marl.. manufariun•r, or olh<•r"i't'. dm•, nol IH'l'l''""il' n111'tilult• 11r impl_1ii'1·ndor,1·nH•nt, H'l'Olllllll'nda1ion. or fa,oring h.' lh<· l nill'd Slall'' CoH·rnm1·nl or the l niH•f'il .' of California. The 1ic"' and upiniun' of aulhof' '' 'P"'"''d IH•rein do nut n1·n·,-.1ril_1 'lat<• or n•nt·c1 lhuw of thl' l ni11•d State' CoH·rnm•·nt lht·n•of. and 'hall nul ht· u"•d for ad•l'rli,ing or producl t•n-dof't'mt•nt purpu'''' · • NOTICE This report was prepared as an account of work sponsored by the United States G overnment. Neither the United States nor the United States Department o f Energy, nor any of their employees, nor any of their contractors, subcontractors, o r their employees, makes any warranty, express or implied, or assumes any lega l liability or responsibility for the accuracy , compl e teness o r usefulness of any information , apparatus, product or process disclosed, or represents that its use would not infringe privately-owned rights. Re ference to a company or pro duct name does not imply approval or recommendation of the product by thr University of Califurnia o r the U.S. Department of Energy to the exclusion of others that may be suitable. Technical Information Department· Lawre:ice Livermore National Laboratory University of California· Livermore, California 94550 . r 

Experimental device for measuring the momentum of disperse granular materials

https://digital.library.unt.edu/ark:/67531/metadc1056019/m2/1/high_res_d/5183372.pdf

Experimental device for measuring the momentum of disperse granular materials

Abstract

Similar works

Full text

Available Versions

UNT Digital Library