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Predictive characterization of particulate solids in the design of storage and transport systems
An Invited Paper- to mark twenty years of publication of the journal Zum 20. Jahresrae dieser Zeitschrift erbetenes Manuskri~t
61
Predictive Characterization of Particulate Solids in the Design of Storage and Transport Systems ~h~akte~sie~~g von S~h~ttg~t~~ bei der Auslegung von Anlagen fiir Lagerung und Transport 0. MOLERUS Lehrstuhl @r Mec~nische Verfahr~steehn~k strasse 9,X520 Erlangen [F.R.G. )
der ~n~versit~t Er~angen-N~~berg,
Murte~-
B. EGERER Knauff Research-Cottrell, Aleed Nobel-Strasse 20, 8700 Wrzburg [FIR. G.) (Received September 4, 1985)
Abstract From the interrelations between material properties and predictions to be made, measurement procedures using direct shear testers are deduced. Measurement accuracy is discussed by comparison with results obtained from more accurate devices. Prediction accuracy is checked by comparison with critical outlet dimensions observed directly from mass flow bins.
Kurzfassung Aus dem Zusammenhang zwischen Materialeigenschaften und den ftir Auslegungszwecke benijtigten Vorhersagen werden die mittels sogenannten direkten Scherme~ger~ten durchzuf~hrenden Messprozeduren abgeleitet. Die Genauigkeit der Messungen wird durch Vergleich mit Resultaten diskutiert, die mit Hilfe genauerer Messanordnungen erzielt wurden. Die Vorhersagegenauigkeit wird durch Vergleich mit unmittelbar gemessenen kritischen Austrittsabmessungen von Massenflussbunkern iiberpriift.
Synopse Die messtechnische Charakterisierung von Schtirt@tern zur Vorhersage des Betriebsverhaltens van Produktionseinrichtungen muss zwei Punkte erfillen. (a) Das Messger& muss zur Sedition der im technischen Apparat ablaufenden Vorgtinge geeignet sein. (b) Die Genauigkeit der Vorhersage basiert auf Reproduzierbarkeit der Messungen und der Eignung der Auswertungsprozedur der prim&en Messdaten fiir die gestellte Aufgabe. Ftir einen elastischen Festkiirper oder ein laminar s~~rne~es Newtonsches Fluid ist das Mater~lv~~~ten durch Stoffkonstanten, ndmlich Elastizitiitsmodui E und Poisso~ah~ v bzw. Viskositdt p festgelegt.
*Paper presented Flow of Particulate 025.52701/86/%3.50
at the International Symposium on Reliable Solids, Bergen, Norway, 20-22 August 1985. Chem. Eng, Process,
Feinkiirnige und damit in der Reel kok&ve Schiittgtiter zeigen dagegen ein sehr vie1 kompiizierteres Mate~~v~halten. Abmessungen und Aus~ass~ffnung eines zylindrixhen Gef&sses k&men so gewrihlt werden, dass ein Teil des Materials ausfliesst, der wandnahe Bereich dagegen in Ruhe bleibt. Verfestigt man das Schiittgut im GefBs durch die Presskraft eines Kolbens, so kann das vorher wenigstens zu Teiiberei~hen ftiessfdhige Material YtiUig unbeweglich gemacht werden. Das Mate~lver~~ten eines koh~s~ven ~h~tt~tes h&t dnher von dessen Belastungsvorgeschichte ab und ist somit nicht einfuch quamiyizierbar. Die physi~lische Ursa&e fiir dieses komplexe Materialverhalten ht in der relntiven Gr&se der zwischen den Partikeln iibertragbaren ffaft~~fte im Vergleich zu den eingepnigten &sseren Kniften und in der Abh&igkeit der Griisse dieser Haftkrfte von den riirsseren ~esskr~ften zu suchen. Physikulische Ursachen fiir Haftkrtifte zwischen Partikeln kiinnen sein: Fh%sigkeitsbrUcken, Festkiirper-
20 (1986) 61-72
0 Elsevier Sequoia/Printed
in The Netherlands
62
briicken durch Kristallisation beim Trocknen, Anlosen van Partikelmaterial, bei feinkornigen Schtittgiitern: van der Waals-Krafte, die mit durch aixserc Pressung hervorgerufene Kontaktdeformation stark zunehmen. Bei gegebener Belastung z. B. durch E2gengewicht sind die Presskrafte proportional zum Partikeldurchmesser im Quadrat, dumit such die Haftkrdfte abhangig von der Partikelgrosse. Aus diesen physikalischen Ursachen der Haftkrafte folgt. (a) Zwei Schiittgiiter zeigen dann dasselbe Materialverhalten wie sie hinsichtlich stofflicher Eigenschaften, Kornverteilung und Gasatmosphare in der Schuttung (relative Feuchte z.B.j ubereinstimmen. (6) Da die genannten Haftkraftmechanismen teilweise Zeitverhalten aufweisen, muss i.a. Zeitabhdngigkeit des Fliessverhaltens eines Schuttgutes erwartet werden. Aus (u) folgt unmittelbar, dass fihesseigenschaften kohasiver Schiittgiiter van Fall zu Fall und insbesondere unter den im Retrieh zu erwartenden Bedingungen zu messen sind. Aus (h) folgt unmittelbar, dass die Messapparatur erkzuben muss, eine Probe zuruichst zu beansptuchen, danach unter Last auszulagern und gegebenenfalls nach mehreren Tagen bei erneutem Einsetzen in die Apparatur die Grenzbedingungen fiir beginnendes Fliessen zu messen. Diese Forderungen verbieten aufwendige Messgerdte, insbesondere s&he, bei denen Messzelle und Messwertaufnehmerfest miteinander verbunden sind. Von den diese Forderungen erfiiNenden Messgerci’ten hat die sog. Jenike-Scherzelle in urspni’nglicher oder vom Benutzer modifiziertw Form die grosste Verbreitung gefimden. Hauptanwendungsgebiet der Jenihe-Scherzelle war und ist die sichere Vorhersage von kn’tischen Mindestaustrittsquerschnitten bei Massenflussbunkern. In diesem Anwendungsfall ergeben sich hinsichtlich der prinzipiell mtiglichen ?‘ierfalt der Beanspruchung,svorgeschichten drastische Vereinfachungen. (a) Bei einem ftrnktionierenden Massenflusshunker war die Beanspruchungsvorgeschichte stets station&es Fhessen des Bunkerinhalts. Stationares Fhessen erfolgt aber derart, dass fortgesetzt Partikelkontakte geliist und neue gebildet werden, d.h. die Haftkrafte sind von den momentan wirkenden ausseren Kraften erzeugt. .4us dieser einfachen fiberlegung folgt, dass in einer o,r-Ebene station&es Hiessen durch einen einzigen stationriren Fliessort darstellbar sein muss. (6) Der Schuttgutaustrag erfolgt in der Praxis iiber Austragsorgane wie Schwingforderer, F~Yrderschnecken, Drehteller usw., und ist daher nicht durch die Wechselwirkung zwischen Schwerkraft und innerer Reibung des Schuttgu tes festgelegt, d.h. Zusammenhange zwischen Spannungszustand und &stand der Deformationsgeschwindigkeit brauchen nicht quantifiziert werden und der Spannungszustand im ruhenden Schr2ttgut nach Schliessen des Austrittsquerschnitts ist gleich dem beim vorangegangenen station&en Fliessen. Aus (a} und (b) folgt unmittelbardie fiirdie Auslegung von Massenflussbunkern geeignete Abfolge der mit der Jenike-Scherzellc durchzufuhrenden Prozeduren: (a) Verfestigen der Sch.&@&probe mit Anscheren bis zum stationaren Fliessen,
(b) zur Ermittlung der Zeitfestigkeit Auslagern der Probe in einer Belastungsbank unter gleichem Lastniveau wie beim Anscheren und (c) fur je eine gem&s (a) und (b) eindeutig festgelegte Variation der Beanspruchungsvorgeschichte durch Normalspannung Ausmesser eines jur beginnendes Fiiessen charakteristischen individuellen iqliessortes 7 = r(of. Auf diese Weise ist je einer bestinzmten Reanspmchungsvorgeschichte, dh. einem Wertepaar o,,, r, auf dem station&en F’liessort eindeutig em individueller Fliessort zugeordnet. Aus den primaren ll/essdaten, d.h. dem station&en Fliessort und den individuellen Fliessorten werden folgende fir die Bunkerauslegung wesentlichen Daten als Funktion der grksten Hauptspannung u1 beim Verfestigen ermittelt: (a) effektivrr Reibungswinkel 6, (b) Druckfestigkeit f,, (c) Zeitdruckjestigkeit fCcL,(11) Druckfestigkeit bei stationarem Fliessen .f,, , (e) Schuttdichte p, (f) Wandreibungswinkel @,, und (bei der Auslegung van Kernflussbunkern) fg) innerer Reibungswinkel I. Zusammen mit theoretisch ermittelten, aus Auslegungsdiagrammen abgelesenen Daten lassen sich dann zwei kritische Austrittsdurchmesser fur vorgegeberre Bunkergeometrie ermitteln: (a) ein grosserer fur Beginn der Bewegung aus der Ruhe und (b) ein kleinerer fur das Erliegen der Rewegung, falls vorher z. B. durch kurzzcitiges Beluften der Fluss zwangsweise in Gang gesetz t war. Spannungszustande werden in einer o,r-Ebene durch Mohrkreise dargestellt. Direkte Schergerate wie das Jenikegerat messen dagegen nur ein Wertepaar cr,r. Vergleichsversuche mit indirekten Schergeraten, die den Spannungszustand vollstandig erfassen, ergaben gute Genauigkeit der mit dem Jenikegerat gewonnenen Duten, Schliesslich ergab der Vergleich zwischen T’orhersage und unmittelbar an einem Bunker genlessenen kritischen Austrittsquerschnitten ebenfalls gute i”berc~ii~stir~lmung.g.
1. Introduction Predictive characterization of particulate solids in the design of storage and transport systems means the supply of data to select the appropriate type of equipment, to predict its mode of operation and to guarantee its reliable functioning. Given the problem, the following requirements concerning measurement devices and procedures are obvious. (i) The measurement device must be suited to simulate operating conditions of the apparatus in question. (ii) Prediction accuracy requires reproducibility of measurement results as well as suitability of the evaluation scheme of the primar-y measurement data. As will be shown. requirements (i) and (ii) are. to some extent, in conflict, thus demanding a compromise. In the following, properties of cohesive powders and their physical origin are discussed first. Then the predictions to be made are presented. From the interrelations between the material properties and the predictions, measurement procedures using direct shear testers are
63 deduced. Reproducibility and comparison of the results of the measurement with those obtained from more accurate devices are discussed. Finally, predictions and directly observed critical outlet dimensions of mass flow bins are compared.
2. Material behavior Prior to investigation of complex material behavior it makes sense to discuss simple material behavior. Such an approach provides a better understanding why, with complex material behavior, only limited goals are pursued with a reasonable chance of success. As prototypes of simple material behavior, a linear elastic material and an incompressible Newtonian liquid are considered (Fig. 1). An elastic rod of length L and square cross-section of area a’, loaded by an axial force F, is elongated by AL and shrinks Aa in the lateral direction. After taking away the load, the rod resumes its initial shape. A second loading by the same force F results in the same deformation as before. As is well known, by introduction of the stress u = F/a2 and the strains en = AL/L and eo = Aa/a, the material is appropriately defined by EL = D/E and by ho = -EL, that is, by two constants, namely Young’s modulus E and the Poisson ratio v. With efflux of a Newtonian liquid out of a large vessel through a pipe of sufficiently small diameter, that is, with laminar flow, the volumetric flow rate Q and pressure drop Ap over the pipe length L are related by Q=
nApd4 128/L
Again, the material behavior can be summarized by a constant, namely the viscosity p. Intermediate pressurization of the total system (application of pressure p) does not change the relation between the pressure drop Ap and the volumetric flow rate Q observed later on. Obviously, the common feature of both types of material behavior is no dependence on loading history
Fii. 1. Simple material incompressible Newtonian
behavior: liquid.
linear elastic
material
(intermediate release of the tensile force and pressurization, respectively). With fine-grained and therefore, in general, more or less cohesive particulate bulk materials a much more complicated behavior is observed (Fig. 2). The relevant dimensions of a container can be chosen such that after opening of an outlet only the central core region is emptied, and an annular zone of stagnant material is preserved (Fig. 2(a)). After compaction of the bulk material by pressing a piston against the top layer (Fig. 2(b)) outflow may be completely inhibited when the outlet is reopened (Fig. 2(c)). lr
b
c
Fig. 2. Material behavior: cohesive particulate material.
In this paper, two materials are defined to be identical if they exhibit the same mechanical behavior during the same loading history; from the simple observations depicted in Fig. 2 the following questions arise. (i) What does ‘identical materials’ mean? (ii) One material, stored in a container, may behave partly as a liquid, partly as a solid (Fig. 2(a)); what does ‘material behavior’ mean in this case? (iii) How should the strong dependence of material behavior on loading history (transition from Fig. 2(a) to (b) and (c)) be quantified?
3. Dependence pressing
of adhesive forces on previous
For a better understanding of the behavior of cohesive materials their real structure has to be taken into account; this consists of a large number of contacting particles. Such a particle approach is unavoidable if one intends to understand, for example, the well-known fact that a solid material, coarse-grained, with an average particle size of about 1 mm in diameter, may behave as cohesionless, whereas the same material, fine-grained, with an average particle diameter of about 10 pm, exhibits strong cohesion effects. Since, in practice, both size distributions result from comminution processes, it can be assumed that in both cases the microstructure of the particle surfaces is roughly the same. The property which changes by orders of magnitude with transition from coarse-grained to fine-grained particles is the load which is transmitted per particle contact. In this fact must be found the dependence of cohesion on particle size. For a given level of stress and given coordination number, the magnitude of the force transmitted per particle contact is
and
Fad2
(1)
64
that is, proportional to the square of the particle size. Relation (1) shows that transition from a particle size of about rl= 10 ym to that of about ci = 1 mm increases the force of pressure transmitted in a particle contact by a factor of 1 04. However, decisive in cohesion is uot the absolute magnitude of the adhesive forces but their relative magnitude in comparison with the forces transmitted in particle contacts resulting from the bulk weight or from external forces. This property of cohesive materials can be explained by results of measurements reported by Schtitz and Schubert [l]. In their investigations, these authors pressed single limestone particles of diameter about 60 pm on metal surfaces in a centrifugal field and then centrifuged them. Their results are represented in Fig. 3 as measured adhesive forces H, against the applied pr-essing force Fp.
effect on the magnitude of the adhesive forces between particles: (1) liquid bridges between particles or capillary liquid inside a powder in bulk as a consequence of wetting, or of condensation of liquid inside a powder in bulk; solid bridges as a consequence of solid material crystallized from solution during drying of powders in bulk; (2) with soluble solid materials, solid bridges from the particle material itself, due to moisture at the contact areas; (3) with elevated temperatures or with solid materials having a low melting point, sintering at the particles contacts (occasionally for-ces during the flow of powders in bulk arc strongly increased by electrostatic effects). Under normal environmental conditions, in particular with fine-grained materials, van der Waals adhesive forces play the dominant role. whereby direct solidsolid interaction is modified by adsorption layers at the solid materral surface, and irreversible contact deformation defines the magnitude of the contact forces observed; that is, soft materials, in general, exhibit stronger adhesive forces than hard materials.
5. Consequences
Fig. 3. Experimental results reported
[ 1 ] cm the dependenceof adhesive
by Schlitz and Schubert force IfD on previous
pressing force PP.
As the log-log representation of the results of the measurements reveals. the adhesive force increases with increasing pressing force relatively strongly at the beginning, whereas further increase in pressing force does not r-e&t in a corresponding increase in adhesive force. The ratio adhesive force to pressing force (HP/F,) which is also drawn in Fig. 3 decreases with increasing pressing for-ce Fp. As a possihlc explanation of this observation, it can be assumed that with pressing of the particles an irreversible deformation of the immediate contact area takes of the true place, which results in a magnification contact area, and therefore in an increase of the van der Waals forces transmitted between particles and substrate. With increasing pressing force the local surface roughness of the particles is increasingly flattened, so that increased pressing does not result in a corresponding increase of true contact area.
4. Physical reasons for adhesive forces between particles According the following
to a representation given by Rumpf [2] different mechanisms can have a decisive
for the design of experiments
The consideration and evaluation of measurements described previously answer the first of the three questions of 53. Figure 3 shows the dependence of the adhesive force on pressing force for a given load level. Two cohesive materials, therefore, can be treated as being identical with respect to their flow behavior if they are equal not only with respect to their material composition. but also with respect to the siLe distribution and shape of their particles, and with respect to the gaseous atmotemperature and sphere inside the bulk, including relative humidity, Therefore, for cohesive materials it is not sufficient to measure Young’s modulus and the Poisson ratio for a certain type of elastic material or the temperaturedependent viscosity for a Newtonian fluid once and for all and then with these to calculate universal material data. f:or a given cohesive material, its flow properties usually have to he measured. The magnitude of the adhesive forces depends on previous pressing (compare Fig. 3). Since, therefore, the flow behavior of a cohesive material dcpcnds on its previous loading history, design of measurement devices as well as the realization of experiments need particular care. As a result of the limited variability in a given measurement device, arbitrary loading histories cannot actually be realized. but only a strictly limited selection. Predictive measurements are used in the design of technical apparatus. Therefore, care has to be taken that a particular measuring device correctly simulates the procedure in the technical apparatus. This demand must not only be met by the design of the measuring device itself but, by strictly agreed rules, also by the execution of the measurement procedure. When results of measurements are compared, it finally has to be proved that the loading histories were also identical.
6.5 Since the proceed with of a material has also to be
majority of the processes described in $4 time, the time dependence of the behavior kept at rest over a distinct time interval taken into account.
6. Common features observed with storage of particulate materials which are relevant to measurement procedures In view of the complexities described above, ageneral, that is, a once-and-for-all, description of a cohesive particulate material is not feasible. Instead, the procedure must be adjusted to the task in question. With the storage of cohesive materials in the silos of industrial plants, the following properties may,in general, be observed: (i) from the plant operating conditions, the time periods over which the silo content is kept at rest (weekends, for example) can be estimated ; (ii) outflow of material is controlled by vibrating feeders, screw feeders, rotating table feeders, or similar devices. From (i) it follows that a device which is suitable for the measurement of those mechanical properties of a powder which are relevant to storage conditions must allow for the following three sequential steps if timedependent consolidation of the powder has to be expected. Firstly, a stress-deformation state has to be established, which corresponds to previous steady state flow of the material. Secondly, time-dependent consolidation under loading conditions prevailing in the silo must be simulated. Thirdly, measurement of the critical conditions at the beginning of flow with reopening of the outlet of the silo is the objective of the whole procedure. Since economic reasons prohibit the use of a measurement device for a single material sample over a period longer than that of the three previously defined steps, the powder sample has to be .confined in a shear cell; this allows for loading and deformation of a powder sample in the measurement device, removal of the shear cell from the measurement device, keeping it under load for a given period of time, and reinserting it into the measurement device for further loading and deformation. This necessary procedure does not allow for sophisticated designs of the shear cell which, in particular,should not be firmly connected to stress or strain recording devices. Fortunately, some essential simplifications concerning the measurement procedure result from (ii). Slow motion of the bulk material is observed with withdrawal of material controlled by feeding devices. Thus, inertial effects are negligible within reasonable limits of accuracy. Furthermore, the outflow is not controlled by the sole interaction of gravitational forces and internal friction of the material. Thus, relations between stresses and deformation rates need not be quantified. However, as shown later on, observation of the stress-strain behavior is absolutely necessary for the control of the adequacy of the measurement procedure itself. Steady state deformation can obviously take place only by continuous loosening of momentary contacts.
Steady state yielding of a powder buIk therefore proceeds in such a way that breakdown of adhesion is just occurring at points of contact due to the interaction between external forces transmitted there and instantaneous cohesive forces generated by these external forces. Thus, for the state of steady state flow attained, a unique relation, that is, a single steady state yield locus, characterizes a particular material. This conclusion obviously leads to a reasonable measurement procedure. With a well-operating silo steady state flow during the last withdrawal of material defines the loading history. Owing to the absence of significant inertial effects, it is reasonable to assume that the same state of stress prevails in the material at rest after closure of the outlet. 7. The Jenike shear testing technique Without doubt, the most widespread and versatile equipment for measuring the shear strength of particulate materials is the Jenike shear cell, The equipment as well as the related measurement procedures have been developed for the purpose of measuring material properties which are relevant to the design of storage silos for unobstructed flow. These properties, which are partly defined within the frame of the radial stress field assumptions [3], are: (i) effective angle of friction 6, (ii) unconfined yield strength f,, (iii) time unconfined yield strength f&, (iv) steady state unconfined yield strength f,,, (v) bulk density p, (vi) kinematic angle of friction between particulate material and wall, I$,. The properties (i)-(vi) are necessary for the design of mass flow bins, whereas (vii) the kinematic angle of internal friction @i must be determined for the design of funnel flow bins. The Jenike shear tester consists of a shear cell (Fig. 4), a weight hanger with weights for the application of vertical pressure, and a mechanism that drives a stem for the application of shear stress. The original Jenike mechanism as well as non-commercial variants developed in different laboratories drive the stem at arate measured in mm min-‘. The driving mechanism measures the force acting on the stem. Again, non-commercialmodifications of the measurement recording device have been developed. The shear cell itself consists of a base, ring, shear cover, mould and a twisting top. The stem acts on the bracket of the shear cover. The pin of the cover transfers a part of the shear force to the ring. First we assume that the cell is filled with a particulate solid which is compacted to a certain degree by a vertical load. The material is then scraped level with the top of the ring. The shear cover is placed on the levelled material and a weight is applied to the shear cover. Then a horizontal force is applied to the bracket by means of the steadily advancing measuring stem. During the shearing operation the ring moves from the original off-set position (Fig. 5(a)) to the off-set position shown in Fig. S(b). The ring thus traverses only a distance equal to the sum of the thicknesses of the wall of the base and the ring. This shortcoming of the Jenike shear cell is the price which has to be paid for a removable shear cell.
66
Fig. 4. Jenike shear tester.
h
(1
Fig. 5. Extreme tion.
positions
of the ring during
the shearing
opera-
.--.____- -7-== /I
Fig. 6. Types of shear force-shearing
distance
curves.
Depending upon the degree of previous compaction, three types of shear force-shearing distance curves are observed (Fig. 6). If the degree of compaction is too small the shear force rises during the whole test (curve a). Such a sample is underconsolidated and the bulk density increases during shear. If the degree of compaction is large, the shear force r-ises at the beginning, passes through a maximum and decreases (curve b). Such a sample is overconsolidated and the bulk density decreases
during shear. However, there exists a distinct degree of compaction when the shear force rises at the beginning, but after having reached a certain value the shear force remains more or less constant during further shear (curve c). Such a sample is said to be critically consolidated. Furthermore, that part of the test in which the shear force is constant corresponds to steady state flow. During steady state flow the bulk densitv remains constant. As indicated by the broken lines, with a shear cell with sufficient shearing distance, as, for example, an annular shear cell, no problem would arise because all three curves would finally attain steady state shear. An annular shear tester, however. is not suited for easy removal of the shear cell and its transfer to a timedependent consolidating station. In the Jenike shear cell, the shearing distance available is usually only 6 mm for the standar-d 95 mm shear cell. Therefore steady state shear must be attained within a shearing distance of about 4 mm, leaving the remaining distance for the actual shear test. As a substitute for sufficient shearing distance, Jenike [3] split the consolidating procedure into two steps. twisting and preshear. The arrangement of the Jenike shear cell for twisting is shown in Fig. 4. The mould is placed on the ring. The cell is filled with the particulate solid; excess material is scraped off level with the top of the mould and a twisting top with a smooth bottom surface is placed on the material. Then the twisting wrench and the hanger with appropriate weights are applied. A certain number of twists is performed (Jenike recommends 30 cycles with 60” amplitudes in both directions). During twisting, the ring must be kept in the off-set position which is defined by locating screws. In order to improve reproducibility of the procedure, noncommercial twisting devices have been developed which ensure centring of the twisting motion as well as absence of additional vertical forces (Fig. 7). After the twisting operation the weight hanger with weights, twisting wrench, mould and twisting top are removed and the material is scraped level with the ring. For- the following preshearing operation. the shear cover is placed on the levelted material and a selected weight is applied by means of the weight hanger. A shear test is performed for the whole shearing distance and the shear force -shearing distance plot is inspected. From this two-stage consolidating procedure, it is obvious that the desired result, that is, a critically consolidated sample according to curve c of Fig. 6, would be attained in the first attempt only by chance. Thus. a trial-and-error procedure, euphemistically called optimization, has to be performed as follows. If the inspection reveals an underconsolidated sample (curve a of Fig. 6). it is necessary to increase the weight during twisting. If the sample is overconsolidated (curve b of Fig. 6) the weight should be reduced during twisting and/or the number of twists decreased. Thus, by repetition, the twisting operation is adjusted for a given preshear operation (given weight) until the shear force--shearing distance plot indicates a critically consolidated sample (curve c in Fig. 6). The crucial point of any shear test consists of the pr-eparation of a critically consolidated sample. From the
67 Division of the weight loads and of the measured shearing forces by the cross-sectional area of the shear cell gives normal stress u and shear stress r transmitted in the horizontal plane which separates the ring and base. Investigation of a material usually means preshear performed at three to four normal stress levels, and for each preshear normal stress level three to five shear normal stress levels measured. The results of such a test can be plotted in a U,T plane as individual yield loci (Yl, YZ, Us in Fig. 9). With reasonable accuracy, the
Fig. 9. Individual
Fig. 7. Twisting
device.
procedure described, it is understandable that shear tests are best performed by persons who understand ‘why something happens’, so technicians who are charged with the performance of shear tests need careful instruction and occasional supervision, particularly if new materials are investigated. The rest of the shear test is a rather routine procedure. For a given weight at preshear a critically consolidated sample has been prepared under the expenditure of about two-thirds of the total shearing distance available. As soon as steady state shear is (approximately) attained, the force measuring stem is retracted, causing the shear stress to fall to zero (Fig. 8). The weight W, of the preshear operation is replaced by the weight W, < W, of the shearing operation and the measuring stem is again switched to the forward direction. As the stem touches the bracket, the shear force rapidly increases, goes through a maximum (representing the yield strength) and starts decreasing.
dlstonce
Fig. 8. Measurement
d-
of a point of the yield locus.
yield loci and steady state yield locus.
individual yield loci can be regarded as straight lines if forbidden points, that is, preshear points and points corresponding to a low load level, are excluded. The points P1, PZ, Ps in Fig. 9 define the steady states of yield attained during preshear. On the assumption that, with reduced normal load, yield loci, i.e. envelopes of Mohr semi-circles, have really been measured, the following evaluation scheme can be applied. That Mohr circle which defines preshear (and thus steady state flow) passes the respective point Pi and is tangential to the individual yield locus Vi. This is a unique rule for drawing the Mohr semi-circles M1, Mz, Ms. The major principal stress u1 during consolidation, that is, during preshear, is then defined by the righthand intersection of the Mohr circle and the abscissa. The steady state yield locus Y,, is then reasonably defined as an envelope of the Mohr circles M1, Mz, MB. As depicted in Fig. 9, evaluation of measurements, in general, yields a straight line for the steady state yield locus which crosses the ordinate above the origin. Taking this result into account, the second and third of the three questions of 5 2 can be answered as follows. Within the scope of storage of particulate solids, material behavior is defined by a unique relation between shear stress and normal stress which characterizes steady state flow (steady state yield locus) and by a multiplicity of relations between shear stress and normal stress (individual yield loci) which define incipient yield. Each of these individual yield loci is uniquely related to one particulate loading history, namely to one Mohr circle of the steady state yield locus. It is convenient to supplement the yield loci measurements by the measurement of the dependence of the bulk density on the major consolidating principal stress ul. The bulk density can be simply determined by weighing base, ring and shear cover with the material after a shear test and calculating the bulk density from the mass of the material in the shear cell and the
68 volume of the bulk material in the cell. The bulk density depends on the preshear weight level and hence on err.
8. Time-dependent
during time-dependent consolidation. This procedure obviously can only be performed if, prior to timcdependent consolidation, the instantaneous yield loci have been measured.
consolidation
As described in 55, in general, time effects must be taken into account with cohesive powders. If time effects are significant, the shear cell (base and ring) containing the prcsheared sample is transferred to a time-dependent consolidating station (Fig. 10) after preshear and removal of the loading weights. The sample is put under pressure by means of the weight carrier and a weight I@,. In order to maintain humidity conditions the sample is protected by a rubber cover during time-dependent consolidation. After elapse of the selected time t, the cell is again transferred to the shear tester; a shear cover is placed on the surface of the material and a weight W, is selected and shear is performed in the usual way. With materials which gain strength during time-dependent consolidation, a higher yield strength will be measured, that is, in the u,r plane the yield locus for time-dependent consolidation will lie above the corresponding instantaneous yield locus. In the consolidating bench only a normal load, not a tangential load, can be applied to the cell. Thus, it is not possible to maintain exactly the same state of stress during time-dependent consolidation as with preshear (states of stress coincide only if the orientations of the principal stresses coincide too). In order to achieve the best approximation, the following procedure is proposed (compare Fig. 9). As described in detail, the end Mohr circles Mr , M2. Ma can be drawn. Thus the major principal stress ur during preshear for a given vertical normal stress up can be read off as depicted in Fig. 9. If a wfeight IV, is selected for time-dependent consolidation for a given weight W, during preshear corresponding to the relation
not the orientation of the principal stresses, but at least the magnitude of the major principal stress, is maintained
9. Wall yield locus The wall yield locus is measured for the range of normal stresses which are expected to act at the wall of the hopper. Five to seven points are normally tested. With measurement of the wall yield locus. the base of the shear cell is replaced by a flat sample of the wall material (Fig. 11). Shims are used to adjust the upper surface of the wall sample in the shearing plane of the shear tester. The weights are selected to form a stack so that all values can be determined successively by removing weights off the top of the stack. The largest weight should correspond to the major consolidating stress of the highest yield locus measured. The measurements are represented as wall yield locus, that is, as wall shear stresses r,,, over normal stress u (Fig. 12).
Fig. Il. Set-up for measurement
of wall yield locus
CL
6
0
Fig. 12. Wall yield locus.
10. Evaluation design
Fii. IO. Consolidating
bench station.
of measurements
for hopper
From the primary measurement results, the determination of which has been described up to now, the data needed for hopper design listed at the beginning of $7 can be determined as follows. For each individual yield locus a Mohr circle passing through the origin of the u,r plane and touching the individual yield locus can be drawn (Fig. 9). The appropriate major principal stress defines the unconfined yield strength f,. Thus a pair of values fC,ur can be determined from any individual Mohr circle. Evaluation of all yield loci gives a relation f; =fful) (Fig. 13). The same procedure can be applied to the time yield loci, providing
69
“.
0
cc
Fig. 13. Unconfined yield strength ye and time-dependent uncon-
fined yield strength & vs. major consolidating stress ~1. a relation fCt =f&far) for the time-dependent unconfined yield strength (Fig. 13). As indicated in Fig. 13 by the broken line, the intersections of the experimental curves f,, fCt with a (semi-)theoretical straight line f,theor. finally -give the critical outlet dimensions of a mass flow bin. The theme of the present paper is the predictive characterization of powders, not the prediction procedure itself. Thus this point is not pursued further here. The steady state yield locus Y,, in Fig. 9 crosses the ordinate above the origin of the U,T axes. Thus a steady state unconfined yield strength f,, has to be attributed to the steady state yield locus. The steady state unconfined yield strength is represented by a parallel to the abscissa in Fig. 13. The intersection of r’,, with fi theor provides a critical outlet dimension for the cease of ‘flow if outflow of material has been enforced by additional measures, for example by short-time aeration of the powder mass [4,5]. The previously described unconfined yield strengths f, and fctare decisive for critical outlet dimensions with initiation of flow after reopening of the outlet. As indicated by Fig. 13, in general, the unconfined yield strengths rise from f,, over f,to fct, as correspondingly do the appropriate critical outlet dimensions. The effective angle of friction S is defined as the slope of the tangent at the end Mohr circle passing the origin of the u,r coordinates (see Fig. 9). It has to be kept in mind that this parameter is defined only within the frame of the radial stress field assumptions [3]. (It is used, for example, for the selection of the appropriate diagram from which the (semi-)theoretical parameter Since any end Mohr circle is fcthem. is determined.) characterized by its major principal stress ur, a relation 6 = Gfor)is again established (Fig. 14).
I
dr'l
-! 0
c
0.5
1.0
1.5
2.0
6
[S]
Fig. 14. Effective angle of friction 6 for limestone powder.
In Fig. 14, points designate values which are read off directly from drawn tangents, whereas the full line is calculated from a measured steady state yield locus [6] _ The comparison shows that the steady state yield locus has a physical significance, but not the effective angle of friction 15. (The steady state yield locus and the effective angle of friction coincide only for vanishing steady state unconfined yield strength&,.) The angle of wall friction c#J,,,is determined as follows (see Fig. 1.5). As described previously, evaluation of measurements gives individual yield loci Y and the respective end Mohr circles M with major principal stress ur. The wall yield locus WY intersects the end Mohr circle at the wall point WP, which is significant for converging channels. The slope of the straight line connecting the origin of the u,r axes and the point WP then defines the angle of wall friction &. Evaluation of all individual yield loci again gives a dependence $,= +,for)(Fig. 16).
I
0 0
6
* 6
Fig. 15. Determination of the angle of w’,ll friction 9,.
Fig. 16. Angle of wall friction vs. major consolidating stress.
The angle of internal friction pi of the powder, which is relevant to the design of funnel flow bins, is finally defined by the slope of the individual yield loci (Fig. 9). Thus a weak dependence @i = @i(cr be obtained by evaluation of all yield loci measured.
11. Unconfined
yield strength of moist powders
A sharp decrease of the flowability of particulate materials is observed with increasing moisture content. With lower states of saturation, that is, with relative liquid contents in the pore space up to 30%, liquid bridges prevail, whereas with higher saturations liquid bridges and liquid-filled capillaries do co-exist. With saturations higher than 90% the sharp decrease in strength of agglomerates, for example, indicates the transition to the slurry regime [7]. From this fact, in general during shear tests the moisture content expected in the field should be maintained in the powder sample, for example by rapid handling in order to minimize evaporation effects. Even rather coarse-grained materials with mean particle sizes of about 1 mm, which may behave in a
70
non-cohesive way in the dry state, show significant cohesion with increasing moisture content. III this case, tensile forces transmitted via liquid bridges and/or capillary forces dominate all other bonding effects between adjacent particles. Based on theoretical considerations and on inspection of measurements on the strength of agglomerates, Tomas [8] derived an equation for the prediction of the unconfined yield strength of moist particulate solids. After a slight modification [h] (more realistic modelling of the uniaxial tensile strength). this equation becomes X.25 sin $i J;=-------1 - sin Qi
(2 -E)(l 4
-E)
li;,X* b---- PP
7 d--
In this equation, r$i designates the angle of internal friction, E the void fraction. ps and pp the solid and liquid density, respectively, xp the liquid/solid mass ratio, y the surface tension of the liquid, and d the (mean) particle size. The equation allows for an estimate of the expected unconfined yield strength without any measurement if reasonable estimates of material data (for example pi Z 30°, e g 0.5) are inserted.
12. Accuracy
of the predictions
In the following, prediction accuracy is discussed with respect to reproducibility, comparison with experimental results obtained from more sophisticated devices, and accuracy of the critical outlet dimensions of bins. The Working Party on the Mechanics of Particulate Solids (WPMPS) of the European Federation of Chemical Engineering investigated the reproducibility of the Jenike shear testing technique [9]. At first, samples of a standard calcite powder were distributed to over 20 laboratories and then the shear strength was measured on the Jenike shear cell available in each laboratory by the technique employed in the respective laboratory. A comparison of the measurements revealed a certain scatter in the results. Detailed analysis suggested that a standardized testing technique should be available. The final report of the WPMPS will contain detailed instructions written by Jenike supplemented by comments formulated by members of the WPMPS and a descriptive example of a shear test. The present authors intend to present the essentials of the more detailed description in the abovementioned report. Only a few comparisons with experimental results obtained from more sophisticated devices are reported in the literature [ 10, 111. The most severe objection to the usual direct shear testers of translational (i.e. Jenikej, annular or rotational type is the fact that only one pair of normal stress o and shear stress 7 is measurable. States of stress, however, are defined by three principal stresses ul, u2, u3 and their orientations (Fig. 17). With powder flow, only the largest Mohr circle, that IS, the maximum and minimum principal stress, is regarded to be significant. Even accepting this assumption, in a strict sense the usual direct shear tester does not do the job required because a unique construction of
63
6
6,
l3
6,
Fig. 17. Mohr circle representation
of states of stress
[ 101,
a Mohr circle is not possible (Mohr circles M,, M2 in Fig. 17) with only one point in the (r,r plane. Evaluation of measurements obtained from the usual direct shear testers is therefore based on the assumption that yield loci, that is, envelopes of Mohr circles, are measured. From a practical point of view these objections can be ignored if comparative measurements with shear testers which are able to measure the complete state of stress reveal sufficient accuracy for the determination of the relevant design data. in particular for the dependence of the unconfined yield strength .r, on the major consolidating stress u, Without detailed analysis ofthe comparative mcasurcment devices. two results arc presented here. (In any case, more complete information on the state of stress is given by a much more complicated measurement procedure in comparison with that of the Jenike shear tester.) Haaker and Rademacher have presented comparative measurements using a modified triaxial {ester [ 101. As Fig. 1X reveals, tests on bentonite gave practically coincident yield loci. Arthur ef al. [l I] published cornparative measurements using a plane strain tester. Figure 19 shows reasonable agreement in the determination of the dependence & =,f;.(c~~‘)for a calcite powder.
0 0
5
10
Fig. 18. Yield locus for bentonite: tester and Jenike shear cell [ 10 1.
20 OM PO,
15
comparison
-
between
triaxial
l5,IKPol
19. Unconfined yield strength of a calcite powder: comparison between plane strain tester and Jenike shear cell [Ill. Fig.
71
13. Alternatives
to the Jenike shear cell
If a powder exhibits large elastic deformations which exceed the range of shearing distances available at the Jenike shear cell, use of an annular shear tester (Fig. 21) might be inevitable. In all other cases, the practical importance of time-dependent consolidation supports a preference for the original or modified versions of the Jenike shear cell. According to the authors’ opinion, shear test results from torsional shear testers, that is, rotational shear testers with full circular cross-sectional area, are suited for comparative measurements, if changes in flowability, but not design data, are within the scope of interest. In those cases, the simpler handling of the torsional shear tester can be decisive.
Fig. 21. Annular shear tester.
Fig. 20. Large-scale mass flow silo, dimensions in mm
14. Conclusions
[ 5,6].
We can proceed further by ignoring all intermediate steps and simply checking the accuracy of the final result, namely the prediction of the critical outlet dimensions of a bin. The present authors [S, 61 investigated a large-scale mass flow silo (see Fig. 20). Directly measured critical outlet dimensions and predictions obtained with a (modified) Jenike shear tester are summarized in Table 1. The mean particle size of the powder investigated was 24 m. As depicted in Fig. 20, the powder was conveyed pneumatically into the measuring silo. A fine-grained powder, however, degasses only very slowly, that is, it remains in a more or less fluidized state over a long period of time. It is therefore not surprising that the larger critical outlet diameter (30 cm) was only measured after keeping the silo content at rest over a period of more than 4 hours. Earlier reopening always gave the smaller critical outlet diameter (13 cm).
Comparison with directly observed critical outlet dimensions of mass flow bins shows reliable prediction accuracy obtained with the Jenike shear tester if the standardized testing technique is applied.
Nomenclature
z E
2, Ftheor. 7 6 HP L
4P
TABLE 1. Measured and predicted critical outlet dimensions of the mass flow silo (Fig. 20) obtained for a limestone powder
Critical outlet diameter at initiation of flow Critical outlet diameter at cessation of flow
Measured
Predicted
30 cm
32 cm
13 cm
14 cm
Q t WP WS wt x!? ; E EC
-__
-
area, mz particle diameter, m Young’s modulus, N m-* unconfined yield strength, N rn-’ time-dependent unconfined yield strength, N m-* uncon8ned yield strength for doming, N rn-’ steady state unconfined yield strength, N mm2 force, N pressing force, N adhesion force after pressing, N length, m pressure drop, N m-* volumetric flow rate, m3 s-l consolidation time, s weight during preshear, N weight during shear, N weight during time-dependent consolidation, N liquid/solid mass ratio
EL
surface tension, N m-l effective angle of friction void fraction = Aa/a = AL/L
72 I-1
viscosity,
N nl?
V
Poisson’s
ratio
P Pn
PS n 01 oi UP
s
bulk density, kg m-’ liquid density, kg m-’ solid density, kg md3 normal stress, N rCz major consolidating principal stress, N m-2 principal stresses (i = 1,2, 3) preshear normal stress, N me2 shear stress, N me2 angle of internal friction angle of wall friction
References 1 W. Schlitz and H. Schubert, 2 H. Rumpf, Chem:Ing.-Tech.,
Chem.-In&-Tech., 48(1976)567. 30 (1958) 144-156.
3 A. W. Jenike, Storage and flow of solids, Bull. No. 108, University of Utah, 1961. 4 G. G. Enstadt, 1st Eur. Symp. on Stress and Strain Behaviour 1984, Abstr. 50-51. of Particulate Solids, Prague, 5 8. bgerer, Dissertation. Universita’t Erlangen Ndmberg, 1982. 6 0. Molerus, Schiittgutmechanik, Crundlagen und Anwendungen in der Sclliittguttnechanik, Springer, B&in, 1985. 7 H. Schubert, KqGllarif~~ irr pordse!i S~sfemerr, Springer, Berlin, 1982. 8 J. Tomas, F’reiberg. Forschungsh. A, (677) (1983). 9 Working Party on the Mechanics of Particulate Solids (WPMPS). The Jcnikc Shear Testing Technique of Particulate Solids, European Federation of’Chemical Enp’neering Publication Series, to he published. 10 C;. Haaker and F. J. C. Rademacher. Proc. Int. Symp. on Powder Technology, K~toto, 1981, The Society of Powder Technology, Kyoto, 1982, pp. 126-132. 11 _I. K. F. Arthur, 1‘. Dunstan and G. G. Enstadt, 1st Eur. Syrup. on Stress and Strain Behaviour of Particulata Solids, Prague, 1984, Abstr. 8-- F.
61
Predictive Characterization of Particulate Solids in the Design of Storage and Transport Systems ~h~akte~sie~~g von S~h~ttg~t~~ bei der Auslegung von Anlagen fiir Lagerung und Transport 0. MOLERUS Lehrstuhl @r Mec~nische Verfahr~steehn~k strasse 9,X520 Erlangen [F.R.G. )
der ~n~versit~t Er~angen-N~~berg,
Murte~-
B. EGERER Knauff Research-Cottrell, Aleed Nobel-Strasse 20, 8700 Wrzburg [FIR. G.) (Received September 4, 1985)
Abstract From the interrelations between material properties and predictions to be made, measurement procedures using direct shear testers are deduced. Measurement accuracy is discussed by comparison with results obtained from more accurate devices. Prediction accuracy is checked by comparison with critical outlet dimensions observed directly from mass flow bins.
Kurzfassung Aus dem Zusammenhang zwischen Materialeigenschaften und den ftir Auslegungszwecke benijtigten Vorhersagen werden die mittels sogenannten direkten Scherme~ger~ten durchzuf~hrenden Messprozeduren abgeleitet. Die Genauigkeit der Messungen wird durch Vergleich mit Resultaten diskutiert, die mit Hilfe genauerer Messanordnungen erzielt wurden. Die Vorhersagegenauigkeit wird durch Vergleich mit unmittelbar gemessenen kritischen Austrittsabmessungen von Massenflussbunkern iiberpriift.
Synopse Die messtechnische Charakterisierung von Schtirt@tern zur Vorhersage des Betriebsverhaltens van Produktionseinrichtungen muss zwei Punkte erfillen. (a) Das Messger& muss zur Sedition der im technischen Apparat ablaufenden Vorgtinge geeignet sein. (b) Die Genauigkeit der Vorhersage basiert auf Reproduzierbarkeit der Messungen und der Eignung der Auswertungsprozedur der prim&en Messdaten fiir die gestellte Aufgabe. Ftir einen elastischen Festkiirper oder ein laminar s~~rne~es Newtonsches Fluid ist das Mater~lv~~~ten durch Stoffkonstanten, ndmlich Elastizitiitsmodui E und Poisso~ah~ v bzw. Viskositdt p festgelegt.
*Paper presented Flow of Particulate 025.52701/86/%3.50
at the International Symposium on Reliable Solids, Bergen, Norway, 20-22 August 1985. Chem. Eng, Process,
Feinkiirnige und damit in der Reel kok&ve Schiittgtiter zeigen dagegen ein sehr vie1 kompiizierteres Mate~~v~halten. Abmessungen und Aus~ass~ffnung eines zylindrixhen Gef&sses k&men so gewrihlt werden, dass ein Teil des Materials ausfliesst, der wandnahe Bereich dagegen in Ruhe bleibt. Verfestigt man das Schiittgut im GefBs durch die Presskraft eines Kolbens, so kann das vorher wenigstens zu Teiiberei~hen ftiessfdhige Material YtiUig unbeweglich gemacht werden. Das Mate~lver~~ten eines koh~s~ven ~h~tt~tes h&t dnher von dessen Belastungsvorgeschichte ab und ist somit nicht einfuch quamiyizierbar. Die physi~lische Ursa&e fiir dieses komplexe Materialverhalten ht in der relntiven Gr&se der zwischen den Partikeln iibertragbaren ffaft~~fte im Vergleich zu den eingepnigten &sseren Kniften und in der Abh&igkeit der Griisse dieser Haftkrfte von den riirsseren ~esskr~ften zu suchen. Physikulische Ursachen fiir Haftkrtifte zwischen Partikeln kiinnen sein: Fh%sigkeitsbrUcken, Festkiirper-
20 (1986) 61-72
0 Elsevier Sequoia/Printed
in The Netherlands
62
briicken durch Kristallisation beim Trocknen, Anlosen van Partikelmaterial, bei feinkornigen Schtittgiitern: van der Waals-Krafte, die mit durch aixserc Pressung hervorgerufene Kontaktdeformation stark zunehmen. Bei gegebener Belastung z. B. durch E2gengewicht sind die Presskrafte proportional zum Partikeldurchmesser im Quadrat, dumit such die Haftkrdfte abhangig von der Partikelgrosse. Aus diesen physikalischen Ursachen der Haftkrafte folgt. (a) Zwei Schiittgiiter zeigen dann dasselbe Materialverhalten wie sie hinsichtlich stofflicher Eigenschaften, Kornverteilung und Gasatmosphare in der Schuttung (relative Feuchte z.B.j ubereinstimmen. (6) Da die genannten Haftkraftmechanismen teilweise Zeitverhalten aufweisen, muss i.a. Zeitabhdngigkeit des Fliessverhaltens eines Schuttgutes erwartet werden. Aus (u) folgt unmittelbar, dass fihesseigenschaften kohasiver Schiittgiiter van Fall zu Fall und insbesondere unter den im Retrieh zu erwartenden Bedingungen zu messen sind. Aus (h) folgt unmittelbar, dass die Messapparatur erkzuben muss, eine Probe zuruichst zu beansptuchen, danach unter Last auszulagern und gegebenenfalls nach mehreren Tagen bei erneutem Einsetzen in die Apparatur die Grenzbedingungen fiir beginnendes Fliessen zu messen. Diese Forderungen verbieten aufwendige Messgerdte, insbesondere s&he, bei denen Messzelle und Messwertaufnehmerfest miteinander verbunden sind. Von den diese Forderungen erfiiNenden Messgerci’ten hat die sog. Jenike-Scherzelle in urspni’nglicher oder vom Benutzer modifiziertw Form die grosste Verbreitung gefimden. Hauptanwendungsgebiet der Jenihe-Scherzelle war und ist die sichere Vorhersage von kn’tischen Mindestaustrittsquerschnitten bei Massenflussbunkern. In diesem Anwendungsfall ergeben sich hinsichtlich der prinzipiell mtiglichen ?‘ierfalt der Beanspruchung,svorgeschichten drastische Vereinfachungen. (a) Bei einem ftrnktionierenden Massenflusshunker war die Beanspruchungsvorgeschichte stets station&es Fhessen des Bunkerinhalts. Stationares Fhessen erfolgt aber derart, dass fortgesetzt Partikelkontakte geliist und neue gebildet werden, d.h. die Haftkrafte sind von den momentan wirkenden ausseren Kraften erzeugt. .4us dieser einfachen fiberlegung folgt, dass in einer o,r-Ebene station&es Hiessen durch einen einzigen stationriren Fliessort darstellbar sein muss. (6) Der Schuttgutaustrag erfolgt in der Praxis iiber Austragsorgane wie Schwingforderer, F~Yrderschnecken, Drehteller usw., und ist daher nicht durch die Wechselwirkung zwischen Schwerkraft und innerer Reibung des Schuttgu tes festgelegt, d.h. Zusammenhange zwischen Spannungszustand und &stand der Deformationsgeschwindigkeit brauchen nicht quantifiziert werden und der Spannungszustand im ruhenden Schr2ttgut nach Schliessen des Austrittsquerschnitts ist gleich dem beim vorangegangenen station&en Fliessen. Aus (a} und (b) folgt unmittelbardie fiirdie Auslegung von Massenflussbunkern geeignete Abfolge der mit der Jenike-Scherzellc durchzufuhrenden Prozeduren: (a) Verfestigen der Sch.&@&probe mit Anscheren bis zum stationaren Fliessen,
(b) zur Ermittlung der Zeitfestigkeit Auslagern der Probe in einer Belastungsbank unter gleichem Lastniveau wie beim Anscheren und (c) fur je eine gem&s (a) und (b) eindeutig festgelegte Variation der Beanspruchungsvorgeschichte durch Normalspannung Ausmesser eines jur beginnendes Fiiessen charakteristischen individuellen iqliessortes 7 = r(of. Auf diese Weise ist je einer bestinzmten Reanspmchungsvorgeschichte, dh. einem Wertepaar o,,, r, auf dem station&en F’liessort eindeutig em individueller Fliessort zugeordnet. Aus den primaren ll/essdaten, d.h. dem station&en Fliessort und den individuellen Fliessorten werden folgende fir die Bunkerauslegung wesentlichen Daten als Funktion der grksten Hauptspannung u1 beim Verfestigen ermittelt: (a) effektivrr Reibungswinkel 6, (b) Druckfestigkeit f,, (c) Zeitdruckjestigkeit fCcL,(11) Druckfestigkeit bei stationarem Fliessen .f,, , (e) Schuttdichte p, (f) Wandreibungswinkel @,, und (bei der Auslegung van Kernflussbunkern) fg) innerer Reibungswinkel I. Zusammen mit theoretisch ermittelten, aus Auslegungsdiagrammen abgelesenen Daten lassen sich dann zwei kritische Austrittsdurchmesser fur vorgegeberre Bunkergeometrie ermitteln: (a) ein grosserer fur Beginn der Bewegung aus der Ruhe und (b) ein kleinerer fur das Erliegen der Rewegung, falls vorher z. B. durch kurzzcitiges Beluften der Fluss zwangsweise in Gang gesetz t war. Spannungszustande werden in einer o,r-Ebene durch Mohrkreise dargestellt. Direkte Schergerate wie das Jenikegerat messen dagegen nur ein Wertepaar cr,r. Vergleichsversuche mit indirekten Schergeraten, die den Spannungszustand vollstandig erfassen, ergaben gute Genauigkeit der mit dem Jenikegerat gewonnenen Duten, Schliesslich ergab der Vergleich zwischen T’orhersage und unmittelbar an einem Bunker genlessenen kritischen Austrittsquerschnitten ebenfalls gute i”berc~ii~stir~lmung.g.
1. Introduction Predictive characterization of particulate solids in the design of storage and transport systems means the supply of data to select the appropriate type of equipment, to predict its mode of operation and to guarantee its reliable functioning. Given the problem, the following requirements concerning measurement devices and procedures are obvious. (i) The measurement device must be suited to simulate operating conditions of the apparatus in question. (ii) Prediction accuracy requires reproducibility of measurement results as well as suitability of the evaluation scheme of the primar-y measurement data. As will be shown. requirements (i) and (ii) are. to some extent, in conflict, thus demanding a compromise. In the following, properties of cohesive powders and their physical origin are discussed first. Then the predictions to be made are presented. From the interrelations between the material properties and the predictions, measurement procedures using direct shear testers are
63 deduced. Reproducibility and comparison of the results of the measurement with those obtained from more accurate devices are discussed. Finally, predictions and directly observed critical outlet dimensions of mass flow bins are compared.
2. Material behavior Prior to investigation of complex material behavior it makes sense to discuss simple material behavior. Such an approach provides a better understanding why, with complex material behavior, only limited goals are pursued with a reasonable chance of success. As prototypes of simple material behavior, a linear elastic material and an incompressible Newtonian liquid are considered (Fig. 1). An elastic rod of length L and square cross-section of area a’, loaded by an axial force F, is elongated by AL and shrinks Aa in the lateral direction. After taking away the load, the rod resumes its initial shape. A second loading by the same force F results in the same deformation as before. As is well known, by introduction of the stress u = F/a2 and the strains en = AL/L and eo = Aa/a, the material is appropriately defined by EL = D/E and by ho = -EL, that is, by two constants, namely Young’s modulus E and the Poisson ratio v. With efflux of a Newtonian liquid out of a large vessel through a pipe of sufficiently small diameter, that is, with laminar flow, the volumetric flow rate Q and pressure drop Ap over the pipe length L are related by Q=
nApd4 128/L
Again, the material behavior can be summarized by a constant, namely the viscosity p. Intermediate pressurization of the total system (application of pressure p) does not change the relation between the pressure drop Ap and the volumetric flow rate Q observed later on. Obviously, the common feature of both types of material behavior is no dependence on loading history
Fii. 1. Simple material incompressible Newtonian
behavior: liquid.
linear elastic
material
(intermediate release of the tensile force and pressurization, respectively). With fine-grained and therefore, in general, more or less cohesive particulate bulk materials a much more complicated behavior is observed (Fig. 2). The relevant dimensions of a container can be chosen such that after opening of an outlet only the central core region is emptied, and an annular zone of stagnant material is preserved (Fig. 2(a)). After compaction of the bulk material by pressing a piston against the top layer (Fig. 2(b)) outflow may be completely inhibited when the outlet is reopened (Fig. 2(c)). lr
b
c
Fig. 2. Material behavior: cohesive particulate material.
In this paper, two materials are defined to be identical if they exhibit the same mechanical behavior during the same loading history; from the simple observations depicted in Fig. 2 the following questions arise. (i) What does ‘identical materials’ mean? (ii) One material, stored in a container, may behave partly as a liquid, partly as a solid (Fig. 2(a)); what does ‘material behavior’ mean in this case? (iii) How should the strong dependence of material behavior on loading history (transition from Fig. 2(a) to (b) and (c)) be quantified?
3. Dependence pressing
of adhesive forces on previous
For a better understanding of the behavior of cohesive materials their real structure has to be taken into account; this consists of a large number of contacting particles. Such a particle approach is unavoidable if one intends to understand, for example, the well-known fact that a solid material, coarse-grained, with an average particle size of about 1 mm in diameter, may behave as cohesionless, whereas the same material, fine-grained, with an average particle diameter of about 10 pm, exhibits strong cohesion effects. Since, in practice, both size distributions result from comminution processes, it can be assumed that in both cases the microstructure of the particle surfaces is roughly the same. The property which changes by orders of magnitude with transition from coarse-grained to fine-grained particles is the load which is transmitted per particle contact. In this fact must be found the dependence of cohesion on particle size. For a given level of stress and given coordination number, the magnitude of the force transmitted per particle contact is
and
Fad2
(1)
64
that is, proportional to the square of the particle size. Relation (1) shows that transition from a particle size of about rl= 10 ym to that of about ci = 1 mm increases the force of pressure transmitted in a particle contact by a factor of 1 04. However, decisive in cohesion is uot the absolute magnitude of the adhesive forces but their relative magnitude in comparison with the forces transmitted in particle contacts resulting from the bulk weight or from external forces. This property of cohesive materials can be explained by results of measurements reported by Schtitz and Schubert [l]. In their investigations, these authors pressed single limestone particles of diameter about 60 pm on metal surfaces in a centrifugal field and then centrifuged them. Their results are represented in Fig. 3 as measured adhesive forces H, against the applied pr-essing force Fp.
effect on the magnitude of the adhesive forces between particles: (1) liquid bridges between particles or capillary liquid inside a powder in bulk as a consequence of wetting, or of condensation of liquid inside a powder in bulk; solid bridges as a consequence of solid material crystallized from solution during drying of powders in bulk; (2) with soluble solid materials, solid bridges from the particle material itself, due to moisture at the contact areas; (3) with elevated temperatures or with solid materials having a low melting point, sintering at the particles contacts (occasionally for-ces during the flow of powders in bulk arc strongly increased by electrostatic effects). Under normal environmental conditions, in particular with fine-grained materials, van der Waals adhesive forces play the dominant role. whereby direct solidsolid interaction is modified by adsorption layers at the solid materral surface, and irreversible contact deformation defines the magnitude of the contact forces observed; that is, soft materials, in general, exhibit stronger adhesive forces than hard materials.
5. Consequences
Fig. 3. Experimental results reported
[ 1 ] cm the dependenceof adhesive
by Schlitz and Schubert force IfD on previous
pressing force PP.
As the log-log representation of the results of the measurements reveals. the adhesive force increases with increasing pressing force relatively strongly at the beginning, whereas further increase in pressing force does not r-e&t in a corresponding increase in adhesive force. The ratio adhesive force to pressing force (HP/F,) which is also drawn in Fig. 3 decreases with increasing pressing for-ce Fp. As a possihlc explanation of this observation, it can be assumed that with pressing of the particles an irreversible deformation of the immediate contact area takes of the true place, which results in a magnification contact area, and therefore in an increase of the van der Waals forces transmitted between particles and substrate. With increasing pressing force the local surface roughness of the particles is increasingly flattened, so that increased pressing does not result in a corresponding increase of true contact area.
4. Physical reasons for adhesive forces between particles According the following
to a representation given by Rumpf [2] different mechanisms can have a decisive
for the design of experiments
The consideration and evaluation of measurements described previously answer the first of the three questions of 53. Figure 3 shows the dependence of the adhesive force on pressing force for a given load level. Two cohesive materials, therefore, can be treated as being identical with respect to their flow behavior if they are equal not only with respect to their material composition. but also with respect to the siLe distribution and shape of their particles, and with respect to the gaseous atmotemperature and sphere inside the bulk, including relative humidity, Therefore, for cohesive materials it is not sufficient to measure Young’s modulus and the Poisson ratio for a certain type of elastic material or the temperaturedependent viscosity for a Newtonian fluid once and for all and then with these to calculate universal material data. f:or a given cohesive material, its flow properties usually have to he measured. The magnitude of the adhesive forces depends on previous pressing (compare Fig. 3). Since, therefore, the flow behavior of a cohesive material dcpcnds on its previous loading history, design of measurement devices as well as the realization of experiments need particular care. As a result of the limited variability in a given measurement device, arbitrary loading histories cannot actually be realized. but only a strictly limited selection. Predictive measurements are used in the design of technical apparatus. Therefore, care has to be taken that a particular measuring device correctly simulates the procedure in the technical apparatus. This demand must not only be met by the design of the measuring device itself but, by strictly agreed rules, also by the execution of the measurement procedure. When results of measurements are compared, it finally has to be proved that the loading histories were also identical.
6.5 Since the proceed with of a material has also to be
majority of the processes described in $4 time, the time dependence of the behavior kept at rest over a distinct time interval taken into account.
6. Common features observed with storage of particulate materials which are relevant to measurement procedures In view of the complexities described above, ageneral, that is, a once-and-for-all, description of a cohesive particulate material is not feasible. Instead, the procedure must be adjusted to the task in question. With the storage of cohesive materials in the silos of industrial plants, the following properties may,in general, be observed: (i) from the plant operating conditions, the time periods over which the silo content is kept at rest (weekends, for example) can be estimated ; (ii) outflow of material is controlled by vibrating feeders, screw feeders, rotating table feeders, or similar devices. From (i) it follows that a device which is suitable for the measurement of those mechanical properties of a powder which are relevant to storage conditions must allow for the following three sequential steps if timedependent consolidation of the powder has to be expected. Firstly, a stress-deformation state has to be established, which corresponds to previous steady state flow of the material. Secondly, time-dependent consolidation under loading conditions prevailing in the silo must be simulated. Thirdly, measurement of the critical conditions at the beginning of flow with reopening of the outlet of the silo is the objective of the whole procedure. Since economic reasons prohibit the use of a measurement device for a single material sample over a period longer than that of the three previously defined steps, the powder sample has to be .confined in a shear cell; this allows for loading and deformation of a powder sample in the measurement device, removal of the shear cell from the measurement device, keeping it under load for a given period of time, and reinserting it into the measurement device for further loading and deformation. This necessary procedure does not allow for sophisticated designs of the shear cell which, in particular,should not be firmly connected to stress or strain recording devices. Fortunately, some essential simplifications concerning the measurement procedure result from (ii). Slow motion of the bulk material is observed with withdrawal of material controlled by feeding devices. Thus, inertial effects are negligible within reasonable limits of accuracy. Furthermore, the outflow is not controlled by the sole interaction of gravitational forces and internal friction of the material. Thus, relations between stresses and deformation rates need not be quantified. However, as shown later on, observation of the stress-strain behavior is absolutely necessary for the control of the adequacy of the measurement procedure itself. Steady state deformation can obviously take place only by continuous loosening of momentary contacts.
Steady state yielding of a powder buIk therefore proceeds in such a way that breakdown of adhesion is just occurring at points of contact due to the interaction between external forces transmitted there and instantaneous cohesive forces generated by these external forces. Thus, for the state of steady state flow attained, a unique relation, that is, a single steady state yield locus, characterizes a particular material. This conclusion obviously leads to a reasonable measurement procedure. With a well-operating silo steady state flow during the last withdrawal of material defines the loading history. Owing to the absence of significant inertial effects, it is reasonable to assume that the same state of stress prevails in the material at rest after closure of the outlet. 7. The Jenike shear testing technique Without doubt, the most widespread and versatile equipment for measuring the shear strength of particulate materials is the Jenike shear cell, The equipment as well as the related measurement procedures have been developed for the purpose of measuring material properties which are relevant to the design of storage silos for unobstructed flow. These properties, which are partly defined within the frame of the radial stress field assumptions [3], are: (i) effective angle of friction 6, (ii) unconfined yield strength f,, (iii) time unconfined yield strength f&, (iv) steady state unconfined yield strength f,,, (v) bulk density p, (vi) kinematic angle of friction between particulate material and wall, I$,. The properties (i)-(vi) are necessary for the design of mass flow bins, whereas (vii) the kinematic angle of internal friction @i must be determined for the design of funnel flow bins. The Jenike shear tester consists of a shear cell (Fig. 4), a weight hanger with weights for the application of vertical pressure, and a mechanism that drives a stem for the application of shear stress. The original Jenike mechanism as well as non-commercial variants developed in different laboratories drive the stem at arate measured in mm min-‘. The driving mechanism measures the force acting on the stem. Again, non-commercialmodifications of the measurement recording device have been developed. The shear cell itself consists of a base, ring, shear cover, mould and a twisting top. The stem acts on the bracket of the shear cover. The pin of the cover transfers a part of the shear force to the ring. First we assume that the cell is filled with a particulate solid which is compacted to a certain degree by a vertical load. The material is then scraped level with the top of the ring. The shear cover is placed on the levelled material and a weight is applied to the shear cover. Then a horizontal force is applied to the bracket by means of the steadily advancing measuring stem. During the shearing operation the ring moves from the original off-set position (Fig. 5(a)) to the off-set position shown in Fig. S(b). The ring thus traverses only a distance equal to the sum of the thicknesses of the wall of the base and the ring. This shortcoming of the Jenike shear cell is the price which has to be paid for a removable shear cell.
66
Fig. 4. Jenike shear tester.
h
(1
Fig. 5. Extreme tion.
positions
of the ring during
the shearing
opera-
.--.____- -7-== /I
Fig. 6. Types of shear force-shearing
distance
curves.
Depending upon the degree of previous compaction, three types of shear force-shearing distance curves are observed (Fig. 6). If the degree of compaction is too small the shear force rises during the whole test (curve a). Such a sample is underconsolidated and the bulk density increases during shear. If the degree of compaction is large, the shear force r-ises at the beginning, passes through a maximum and decreases (curve b). Such a sample is overconsolidated and the bulk density decreases
during shear. However, there exists a distinct degree of compaction when the shear force rises at the beginning, but after having reached a certain value the shear force remains more or less constant during further shear (curve c). Such a sample is said to be critically consolidated. Furthermore, that part of the test in which the shear force is constant corresponds to steady state flow. During steady state flow the bulk densitv remains constant. As indicated by the broken lines, with a shear cell with sufficient shearing distance, as, for example, an annular shear cell, no problem would arise because all three curves would finally attain steady state shear. An annular shear tester, however. is not suited for easy removal of the shear cell and its transfer to a timedependent consolidating station. In the Jenike shear cell, the shearing distance available is usually only 6 mm for the standar-d 95 mm shear cell. Therefore steady state shear must be attained within a shearing distance of about 4 mm, leaving the remaining distance for the actual shear test. As a substitute for sufficient shearing distance, Jenike [3] split the consolidating procedure into two steps. twisting and preshear. The arrangement of the Jenike shear cell for twisting is shown in Fig. 4. The mould is placed on the ring. The cell is filled with the particulate solid; excess material is scraped off level with the top of the mould and a twisting top with a smooth bottom surface is placed on the material. Then the twisting wrench and the hanger with appropriate weights are applied. A certain number of twists is performed (Jenike recommends 30 cycles with 60” amplitudes in both directions). During twisting, the ring must be kept in the off-set position which is defined by locating screws. In order to improve reproducibility of the procedure, noncommercial twisting devices have been developed which ensure centring of the twisting motion as well as absence of additional vertical forces (Fig. 7). After the twisting operation the weight hanger with weights, twisting wrench, mould and twisting top are removed and the material is scraped level with the ring. For- the following preshearing operation. the shear cover is placed on the levelted material and a selected weight is applied by means of the weight hanger. A shear test is performed for the whole shearing distance and the shear force -shearing distance plot is inspected. From this two-stage consolidating procedure, it is obvious that the desired result, that is, a critically consolidated sample according to curve c of Fig. 6, would be attained in the first attempt only by chance. Thus. a trial-and-error procedure, euphemistically called optimization, has to be performed as follows. If the inspection reveals an underconsolidated sample (curve a of Fig. 6). it is necessary to increase the weight during twisting. If the sample is overconsolidated (curve b of Fig. 6) the weight should be reduced during twisting and/or the number of twists decreased. Thus, by repetition, the twisting operation is adjusted for a given preshear operation (given weight) until the shear force--shearing distance plot indicates a critically consolidated sample (curve c in Fig. 6). The crucial point of any shear test consists of the pr-eparation of a critically consolidated sample. From the
67 Division of the weight loads and of the measured shearing forces by the cross-sectional area of the shear cell gives normal stress u and shear stress r transmitted in the horizontal plane which separates the ring and base. Investigation of a material usually means preshear performed at three to four normal stress levels, and for each preshear normal stress level three to five shear normal stress levels measured. The results of such a test can be plotted in a U,T plane as individual yield loci (Yl, YZ, Us in Fig. 9). With reasonable accuracy, the
Fig. 9. Individual
Fig. 7. Twisting
device.
procedure described, it is understandable that shear tests are best performed by persons who understand ‘why something happens’, so technicians who are charged with the performance of shear tests need careful instruction and occasional supervision, particularly if new materials are investigated. The rest of the shear test is a rather routine procedure. For a given weight at preshear a critically consolidated sample has been prepared under the expenditure of about two-thirds of the total shearing distance available. As soon as steady state shear is (approximately) attained, the force measuring stem is retracted, causing the shear stress to fall to zero (Fig. 8). The weight W, of the preshear operation is replaced by the weight W, < W, of the shearing operation and the measuring stem is again switched to the forward direction. As the stem touches the bracket, the shear force rapidly increases, goes through a maximum (representing the yield strength) and starts decreasing.
dlstonce
Fig. 8. Measurement
d-
of a point of the yield locus.
yield loci and steady state yield locus.
individual yield loci can be regarded as straight lines if forbidden points, that is, preshear points and points corresponding to a low load level, are excluded. The points P1, PZ, Ps in Fig. 9 define the steady states of yield attained during preshear. On the assumption that, with reduced normal load, yield loci, i.e. envelopes of Mohr semi-circles, have really been measured, the following evaluation scheme can be applied. That Mohr circle which defines preshear (and thus steady state flow) passes the respective point Pi and is tangential to the individual yield locus Vi. This is a unique rule for drawing the Mohr semi-circles M1, Mz, Ms. The major principal stress u1 during consolidation, that is, during preshear, is then defined by the righthand intersection of the Mohr circle and the abscissa. The steady state yield locus Y,, is then reasonably defined as an envelope of the Mohr circles M1, Mz, MB. As depicted in Fig. 9, evaluation of measurements, in general, yields a straight line for the steady state yield locus which crosses the ordinate above the origin. Taking this result into account, the second and third of the three questions of 5 2 can be answered as follows. Within the scope of storage of particulate solids, material behavior is defined by a unique relation between shear stress and normal stress which characterizes steady state flow (steady state yield locus) and by a multiplicity of relations between shear stress and normal stress (individual yield loci) which define incipient yield. Each of these individual yield loci is uniquely related to one particulate loading history, namely to one Mohr circle of the steady state yield locus. It is convenient to supplement the yield loci measurements by the measurement of the dependence of the bulk density on the major consolidating principal stress ul. The bulk density can be simply determined by weighing base, ring and shear cover with the material after a shear test and calculating the bulk density from the mass of the material in the shear cell and the
68 volume of the bulk material in the cell. The bulk density depends on the preshear weight level and hence on err.
8. Time-dependent
during time-dependent consolidation. This procedure obviously can only be performed if, prior to timcdependent consolidation, the instantaneous yield loci have been measured.
consolidation
As described in 55, in general, time effects must be taken into account with cohesive powders. If time effects are significant, the shear cell (base and ring) containing the prcsheared sample is transferred to a time-dependent consolidating station (Fig. 10) after preshear and removal of the loading weights. The sample is put under pressure by means of the weight carrier and a weight I@,. In order to maintain humidity conditions the sample is protected by a rubber cover during time-dependent consolidation. After elapse of the selected time t, the cell is again transferred to the shear tester; a shear cover is placed on the surface of the material and a weight W, is selected and shear is performed in the usual way. With materials which gain strength during time-dependent consolidation, a higher yield strength will be measured, that is, in the u,r plane the yield locus for time-dependent consolidation will lie above the corresponding instantaneous yield locus. In the consolidating bench only a normal load, not a tangential load, can be applied to the cell. Thus, it is not possible to maintain exactly the same state of stress during time-dependent consolidation as with preshear (states of stress coincide only if the orientations of the principal stresses coincide too). In order to achieve the best approximation, the following procedure is proposed (compare Fig. 9). As described in detail, the end Mohr circles Mr , M2. Ma can be drawn. Thus the major principal stress ur during preshear for a given vertical normal stress up can be read off as depicted in Fig. 9. If a wfeight IV, is selected for time-dependent consolidation for a given weight W, during preshear corresponding to the relation
not the orientation of the principal stresses, but at least the magnitude of the major principal stress, is maintained
9. Wall yield locus The wall yield locus is measured for the range of normal stresses which are expected to act at the wall of the hopper. Five to seven points are normally tested. With measurement of the wall yield locus. the base of the shear cell is replaced by a flat sample of the wall material (Fig. 11). Shims are used to adjust the upper surface of the wall sample in the shearing plane of the shear tester. The weights are selected to form a stack so that all values can be determined successively by removing weights off the top of the stack. The largest weight should correspond to the major consolidating stress of the highest yield locus measured. The measurements are represented as wall yield locus, that is, as wall shear stresses r,,, over normal stress u (Fig. 12).
Fig. Il. Set-up for measurement
of wall yield locus
CL
6
0
Fig. 12. Wall yield locus.
10. Evaluation design
Fii. IO. Consolidating
bench station.
of measurements
for hopper
From the primary measurement results, the determination of which has been described up to now, the data needed for hopper design listed at the beginning of $7 can be determined as follows. For each individual yield locus a Mohr circle passing through the origin of the u,r plane and touching the individual yield locus can be drawn (Fig. 9). The appropriate major principal stress defines the unconfined yield strength f,. Thus a pair of values fC,ur can be determined from any individual Mohr circle. Evaluation of all yield loci gives a relation f; =fful) (Fig. 13). The same procedure can be applied to the time yield loci, providing
69
“.
0
cc
Fig. 13. Unconfined yield strength ye and time-dependent uncon-
fined yield strength & vs. major consolidating stress ~1. a relation fCt =f&far) for the time-dependent unconfined yield strength (Fig. 13). As indicated in Fig. 13 by the broken line, the intersections of the experimental curves f,, fCt with a (semi-)theoretical straight line f,theor. finally -give the critical outlet dimensions of a mass flow bin. The theme of the present paper is the predictive characterization of powders, not the prediction procedure itself. Thus this point is not pursued further here. The steady state yield locus Y,, in Fig. 9 crosses the ordinate above the origin of the U,T axes. Thus a steady state unconfined yield strength f,, has to be attributed to the steady state yield locus. The steady state unconfined yield strength is represented by a parallel to the abscissa in Fig. 13. The intersection of r’,, with fi theor provides a critical outlet dimension for the cease of ‘flow if outflow of material has been enforced by additional measures, for example by short-time aeration of the powder mass [4,5]. The previously described unconfined yield strengths f, and fctare decisive for critical outlet dimensions with initiation of flow after reopening of the outlet. As indicated by Fig. 13, in general, the unconfined yield strengths rise from f,, over f,to fct, as correspondingly do the appropriate critical outlet dimensions. The effective angle of friction S is defined as the slope of the tangent at the end Mohr circle passing the origin of the u,r coordinates (see Fig. 9). It has to be kept in mind that this parameter is defined only within the frame of the radial stress field assumptions [3]. (It is used, for example, for the selection of the appropriate diagram from which the (semi-)theoretical parameter Since any end Mohr circle is fcthem. is determined.) characterized by its major principal stress ur, a relation 6 = Gfor)is again established (Fig. 14).
I
dr'l
-! 0
c
0.5
1.0
1.5
2.0
6
[S]
Fig. 14. Effective angle of friction 6 for limestone powder.
In Fig. 14, points designate values which are read off directly from drawn tangents, whereas the full line is calculated from a measured steady state yield locus [6] _ The comparison shows that the steady state yield locus has a physical significance, but not the effective angle of friction 15. (The steady state yield locus and the effective angle of friction coincide only for vanishing steady state unconfined yield strength&,.) The angle of wall friction c#J,,,is determined as follows (see Fig. 1.5). As described previously, evaluation of measurements gives individual yield loci Y and the respective end Mohr circles M with major principal stress ur. The wall yield locus WY intersects the end Mohr circle at the wall point WP, which is significant for converging channels. The slope of the straight line connecting the origin of the u,r axes and the point WP then defines the angle of wall friction &. Evaluation of all individual yield loci again gives a dependence $,= +,for)(Fig. 16).
I
0 0
6
* 6
Fig. 15. Determination of the angle of w’,ll friction 9,.
Fig. 16. Angle of wall friction vs. major consolidating stress.
The angle of internal friction pi of the powder, which is relevant to the design of funnel flow bins, is finally defined by the slope of the individual yield loci (Fig. 9). Thus a weak dependence @i = @i(cr be obtained by evaluation of all yield loci measured.
11. Unconfined
yield strength of moist powders
A sharp decrease of the flowability of particulate materials is observed with increasing moisture content. With lower states of saturation, that is, with relative liquid contents in the pore space up to 30%, liquid bridges prevail, whereas with higher saturations liquid bridges and liquid-filled capillaries do co-exist. With saturations higher than 90% the sharp decrease in strength of agglomerates, for example, indicates the transition to the slurry regime [7]. From this fact, in general during shear tests the moisture content expected in the field should be maintained in the powder sample, for example by rapid handling in order to minimize evaporation effects. Even rather coarse-grained materials with mean particle sizes of about 1 mm, which may behave in a
70
non-cohesive way in the dry state, show significant cohesion with increasing moisture content. III this case, tensile forces transmitted via liquid bridges and/or capillary forces dominate all other bonding effects between adjacent particles. Based on theoretical considerations and on inspection of measurements on the strength of agglomerates, Tomas [8] derived an equation for the prediction of the unconfined yield strength of moist particulate solids. After a slight modification [h] (more realistic modelling of the uniaxial tensile strength). this equation becomes X.25 sin $i J;=-------1 - sin Qi
(2 -E)(l 4
-E)
li;,X* b---- PP
7 d--
In this equation, r$i designates the angle of internal friction, E the void fraction. ps and pp the solid and liquid density, respectively, xp the liquid/solid mass ratio, y the surface tension of the liquid, and d the (mean) particle size. The equation allows for an estimate of the expected unconfined yield strength without any measurement if reasonable estimates of material data (for example pi Z 30°, e g 0.5) are inserted.
12. Accuracy
of the predictions
In the following, prediction accuracy is discussed with respect to reproducibility, comparison with experimental results obtained from more sophisticated devices, and accuracy of the critical outlet dimensions of bins. The Working Party on the Mechanics of Particulate Solids (WPMPS) of the European Federation of Chemical Engineering investigated the reproducibility of the Jenike shear testing technique [9]. At first, samples of a standard calcite powder were distributed to over 20 laboratories and then the shear strength was measured on the Jenike shear cell available in each laboratory by the technique employed in the respective laboratory. A comparison of the measurements revealed a certain scatter in the results. Detailed analysis suggested that a standardized testing technique should be available. The final report of the WPMPS will contain detailed instructions written by Jenike supplemented by comments formulated by members of the WPMPS and a descriptive example of a shear test. The present authors intend to present the essentials of the more detailed description in the abovementioned report. Only a few comparisons with experimental results obtained from more sophisticated devices are reported in the literature [ 10, 111. The most severe objection to the usual direct shear testers of translational (i.e. Jenikej, annular or rotational type is the fact that only one pair of normal stress o and shear stress 7 is measurable. States of stress, however, are defined by three principal stresses ul, u2, u3 and their orientations (Fig. 17). With powder flow, only the largest Mohr circle, that IS, the maximum and minimum principal stress, is regarded to be significant. Even accepting this assumption, in a strict sense the usual direct shear tester does not do the job required because a unique construction of
63
6
6,
l3
6,
Fig. 17. Mohr circle representation
of states of stress
[ 101,
a Mohr circle is not possible (Mohr circles M,, M2 in Fig. 17) with only one point in the (r,r plane. Evaluation of measurements obtained from the usual direct shear testers is therefore based on the assumption that yield loci, that is, envelopes of Mohr circles, are measured. From a practical point of view these objections can be ignored if comparative measurements with shear testers which are able to measure the complete state of stress reveal sufficient accuracy for the determination of the relevant design data. in particular for the dependence of the unconfined yield strength .r, on the major consolidating stress u, Without detailed analysis ofthe comparative mcasurcment devices. two results arc presented here. (In any case, more complete information on the state of stress is given by a much more complicated measurement procedure in comparison with that of the Jenike shear tester.) Haaker and Rademacher have presented comparative measurements using a modified triaxial {ester [ 101. As Fig. 1X reveals, tests on bentonite gave practically coincident yield loci. Arthur ef al. [l I] published cornparative measurements using a plane strain tester. Figure 19 shows reasonable agreement in the determination of the dependence & =,f;.(c~~‘)for a calcite powder.
0 0
5
10
Fig. 18. Yield locus for bentonite: tester and Jenike shear cell [ 10 1.
20 OM PO,
15
comparison
-
between
triaxial
l5,IKPol
19. Unconfined yield strength of a calcite powder: comparison between plane strain tester and Jenike shear cell [Ill. Fig.
71
13. Alternatives
to the Jenike shear cell
If a powder exhibits large elastic deformations which exceed the range of shearing distances available at the Jenike shear cell, use of an annular shear tester (Fig. 21) might be inevitable. In all other cases, the practical importance of time-dependent consolidation supports a preference for the original or modified versions of the Jenike shear cell. According to the authors’ opinion, shear test results from torsional shear testers, that is, rotational shear testers with full circular cross-sectional area, are suited for comparative measurements, if changes in flowability, but not design data, are within the scope of interest. In those cases, the simpler handling of the torsional shear tester can be decisive.
Fig. 21. Annular shear tester.
Fig. 20. Large-scale mass flow silo, dimensions in mm
14. Conclusions
[ 5,6].
We can proceed further by ignoring all intermediate steps and simply checking the accuracy of the final result, namely the prediction of the critical outlet dimensions of a bin. The present authors [S, 61 investigated a large-scale mass flow silo (see Fig. 20). Directly measured critical outlet dimensions and predictions obtained with a (modified) Jenike shear tester are summarized in Table 1. The mean particle size of the powder investigated was 24 m. As depicted in Fig. 20, the powder was conveyed pneumatically into the measuring silo. A fine-grained powder, however, degasses only very slowly, that is, it remains in a more or less fluidized state over a long period of time. It is therefore not surprising that the larger critical outlet diameter (30 cm) was only measured after keeping the silo content at rest over a period of more than 4 hours. Earlier reopening always gave the smaller critical outlet diameter (13 cm).
Comparison with directly observed critical outlet dimensions of mass flow bins shows reliable prediction accuracy obtained with the Jenike shear tester if the standardized testing technique is applied.
Nomenclature
z E
2, Ftheor. 7 6 HP L
4P
TABLE 1. Measured and predicted critical outlet dimensions of the mass flow silo (Fig. 20) obtained for a limestone powder
Critical outlet diameter at initiation of flow Critical outlet diameter at cessation of flow
Measured
Predicted
30 cm
32 cm
13 cm
14 cm
Q t WP WS wt x!? ; E EC
-__
-
area, mz particle diameter, m Young’s modulus, N m-* unconfined yield strength, N rn-’ time-dependent unconfined yield strength, N m-* uncon8ned yield strength for doming, N rn-’ steady state unconfined yield strength, N mm2 force, N pressing force, N adhesion force after pressing, N length, m pressure drop, N m-* volumetric flow rate, m3 s-l consolidation time, s weight during preshear, N weight during shear, N weight during time-dependent consolidation, N liquid/solid mass ratio
EL
surface tension, N m-l effective angle of friction void fraction = Aa/a = AL/L
72 I-1
viscosity,
N nl?
V
Poisson’s
ratio
P Pn
PS n 01 oi UP
s
bulk density, kg m-’ liquid density, kg m-’ solid density, kg md3 normal stress, N rCz major consolidating principal stress, N m-2 principal stresses (i = 1,2, 3) preshear normal stress, N me2 shear stress, N me2 angle of internal friction angle of wall friction
References 1 W. Schlitz and H. Schubert, 2 H. Rumpf, Chem:Ing.-Tech.,
Chem.-In&-Tech., 48(1976)567. 30 (1958) 144-156.
3 A. W. Jenike, Storage and flow of solids, Bull. No. 108, University of Utah, 1961. 4 G. G. Enstadt, 1st Eur. Symp. on Stress and Strain Behaviour 1984, Abstr. 50-51. of Particulate Solids, Prague, 5 8. bgerer, Dissertation. Universita’t Erlangen Ndmberg, 1982. 6 0. Molerus, Schiittgutmechanik, Crundlagen und Anwendungen in der Sclliittguttnechanik, Springer, B&in, 1985. 7 H. Schubert, KqGllarif~~ irr pordse!i S~sfemerr, Springer, Berlin, 1982. 8 J. Tomas, F’reiberg. Forschungsh. A, (677) (1983). 9 Working Party on the Mechanics of Particulate Solids (WPMPS). The Jcnikc Shear Testing Technique of Particulate Solids, European Federation of’Chemical Enp’neering Publication Series, to he published. 10 C;. Haaker and F. J. C. Rademacher. Proc. Int. Symp. on Powder Technology, K~toto, 1981, The Society of Powder Technology, Kyoto, 1982, pp. 126-132. 11 _I. K. F. Arthur, 1‘. Dunstan and G. G. Enstadt, 1st Eur. Syrup. on Stress and Strain Behaviour of Particulata Solids, Prague, 1984, Abstr. 8-- F.