Numerical Simulation of Aerated Powder Consolidation · Numerical Simulation of Aerated Powder...
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Numerical Simulation of Aerated Powder Consolidation 1 Kristy A. Coffey and Pierre A. Gremaud Department of Mathematics and Center for Research in Scientific Computation, North Carolina State University, Raleigh, NC 27695-8205, USA When a fine powder is dumped into a silo, the gas trapped by the particles will slowly escape by diffusing through the material. The corresponding uneven gas pressure distribution creates a body force that is taken into account through Darcy’s law. By using spatial averaging, the formulation, even though essentially one-dimensional in space, includes effects due the geometry of the container. An efficient and robust numerical scheme based on a DAE formulation is proposed and implemented. Various computational results are presented and discussed to establish the validity of the approach. 1. INTRODUCTION This paper deals with various problems related to the simulation of aerated powder consolidation. This kind of phenomena is routinely encountered during the handling of fine powders in countless applications. Typically a powder is stored in a bunker or silo, see Figure 1. During filling, air gets trapped in the material leading, in some cases, to partial fluidization [8], [14] and noticeable changes of the mechanical properties. Over time the excess air diffuses through the powder and eventually escapes through the top surface. This paper aims to find the length of time a given material takes to consolidate. With respect to applications, the ultimate goal is to be able to predict and consequently avoid flooding, i.e., the sudden discharge from a hopper of a fine powder at a much greater rate than that of the flow of ordinary granular materials. Further comments on the connection between flooding and deaeration can be found in [12]. Early work by Janssen [6] analyzed the behavior of a column of granular material in a container. Forces due to the gradient of gas pressure were neglected, and the analysis was restricted to vertical cylinders. Some other models are based on drastic simplifications, such as considering a constant vertical stress [9]. The present model is derived from basic conservation principles of mass and momentum. The forces resulting from nonuniform gas pressure are taken into account through Darcy’s law. The modeling assumptions roughly follow that of [8], see also [4], [12]. In those publications, the effects of the geometry of the container are neglected, in effect, treating only the case of cylindrical bunkers. In [4], that case is mathematically analyzed, and a This project was supported by the Army Research Office (ARO) through grant DAAD19-99-1-0188. The first author was partially supported by a Department of Education GAANN Fellowship. The second author was partially supported by the National Science Foundation (NSF) through grant DMS-9818900. 1
Transcript of Numerical Simulation of Aerated Powder Consolidation · Numerical Simulation of Aerated Powder...
Numerical Simulation of Aerated PowderConsolidation1
Kristy A. Coffey andPierreA. GremaudDepartmentof MathematicsandCenterfor Research in ScientificComputation,North Carolina State
University, Raleigh,NC27695-8205,USA
Whenafinepowderisdumpedintoasilo, thegastrappedby theparticleswill slowly escape
by diffusing throughthe material. The correspondinguneven gaspressuredistribution
createsa body force that is taken into accountthroughDarcy’s law. By using spatial
averaging,the formulation, even thoughessentiallyone-dimensionalin space,includes
effectsduethegeometryof thecontainer. An efficient androbustnumericalschemebased
on a DAE formulation is proposedand implemented.Variouscomputationalresultsare
presentedanddiscussedto establishthevalidity of theapproach.
1. INTRODUCTION
This paperdealswith variousproblemsrelatedto the simulationof aeratedpowderconsolidation. This kind of phenomenais routinely encounteredduring the handlingoffinepowdersin countlessapplications.Typically apowderis storedin abunkeror silo, seeFigure1. During filling, air getstrappedin thematerialleading,in somecases,to partialfluidization [8], [14] andnoticeablechangesof themechanicalproperties.Over time theexcessair diffusesthroughthepowderandeventuallyescapesthroughthetopsurface.Thispaperaimsto find the lengthof time a givenmaterialtakesto consolidate.With respectto applications,theultimategoal is to beableto predictandconsequentlyavoid flooding,i.e., thesuddendischargefrom a hopperof a fine powderat a muchgreaterratethanthatof theflow of ordinarygranularmaterials.Furthercommentson theconnectionbetweenfloodinganddeaerationcanbefoundin [12].
Early work by Janssen[6] analyzedthebehavior of a columnof granularmaterialin acontainer. Forcesdueto thegradientof gaspressurewereneglected,andtheanalysiswasrestrictedto vertical cylinders. Someothermodelsarebasedon drasticsimplifications,suchasconsideringa constantverticalstress[9].
Thepresentmodelisderivedfrombasicconservationprinciplesof massandmomentum.Theforcesresultingfrom nonuniformgaspressurearetakeninto accountthroughDarcy’slaw. The modelingassumptionsroughly follow that of [8], seealso [4], [12]. In thosepublications,theeffectsof thegeometryof thecontainerareneglected,in effect, treatingonly the caseof cylindrical bunkers. In [4], that caseis mathematicallyanalyzed,anda�
This projectwassupportedby the Army ResearchOffice (ARO) throughgrantDAAD19-99-1-0188. Thefirst authorwaspartially supportedby a Departmentof EducationGAANN Fellowship. Thesecondauthorwaspartially supportedby theNationalScienceFoundation(NSF)throughgrantDMS-9818900.
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2 K.A. KRISTY AND P.A. GREMAUD
H(t)
z
bunkertop surface
δz
A(z) = R(z)π 2
A(z+ z) = R(z+ z)δ δπ 2
FIG. 1. Geometryandcoordinatesystemsfor the vertical conicalbunker. The heightof the columnofpowderof time
�is denoted��� ��� .
robustnumericalmethodis proposedandimplemented.Oneof the contributionsof thispaperis the extensionof thosemodelsto generalaxisymmetricdomains. Although themodificationmayappearasatechnicaldetail,it hasprofoundramificationswith respecttothestructureof thesystem,seeremarksat theendof � 2.
Apart from using specificphysicalconstitutive laws, the main restrictive assumptionconsistsof neglectingthefluctuationsin thehorizontaldirections,allowingfor anessentiallyonedimensionalin spaceformulation.In otherwords,theproblemis describedexclusivelyin termsof quantitiesthat have beenaveragedin the horizontaldirection. It shouldbenoticedthatin many situationstheuseof quasione-dimensionalconsolidationmodelscanonly beviewedasa first step,andfull multidimensionalapproachesshouldbeconsideredinstead.Wereferto [16] and[10], Chap.5,for remarksaboutthelimitationsof suchmodels.Moreimportantlyhowever, evenasimplifiedmodelsuchastheoneconsideredhereis verysensitive to the valuesof variousmaterialcoefficients suchas compressibility. This isclearly illustratedby our numericalresults,seee.g. Figure5. Thosematerialcoefficientsaretypically hardto measurein anaccurateway.
The paperis organizedasfollows. The model is derived in Section2. The resultingsystemis nonlinearandstronglycoupled. It consistsof anessentiallyparabolicPDE,anODEandanintegralequation.Thethreemainunknownsaretheaverageverticalstress,theaveragegaspressureandtheheightof thepowderin thecontainer. Throughanappropriatetransformation,thecalculationsareperformedin a fixedreferencecomputationaldomain.A discretizationisproposedin Section3. Thespatialdiscretizationissecondorderaccurateandusesacombinationof centeredFiniteDifferencesandBDF[1], [5]. Thesemidiscretizedin spacesystemcorrespondsto a semi-explicit index 2 Differential Algebraic Equation(DAE). The time discretizationis doneby a linearly implicit Eulerdiscretization,which,althoughonly first order, is thesimplestacceptablenumericalapproachfor theabovetypeof DAEs. Our numericalexperimentsshow it to bebothrobustandefficient in thepresentcontext. ComputationalresultsarepresentedanddiscussedSection4. Finally, Section5 isdevotedto concludingremarks.
2. THE MODEL
Weconsidergeneralaxisymmetrichoppersasillustratedin Figure1. In orderto simplifytheproblem,a pseudoone-dimensionalformulationis derivedby averagingall quantitieson horizontalcross-sections.Theheightof thepowder in thecontainerat a generictime�
can thusbe describedby a function � � � . Using cylindrical coordinates,the domain
POWDERCONSOLIDATION 3
occupiedby thepowderat time � is givenby������� ������������� �"!$#�� � �%���"!&�'!$(*)+����!$�,� -"���.�0/1�where
-"���.�standsfor theradiusof thehopperat height
�.
For anarbitraryfunction 2 , we define32 ���.�54 6)7-981���.� :&;=<?>�@A : 8�BA 2 ����� �������.�DC1�DC1���where
32 ���.� is theaveragedvalueof thefunction 2 atheight�
in thehopper. By axisymmetry,thefunctionsconsideredheredonotdependontheangularvariable
�, andthuswith aslight
abuseof notation 32 ���.�D4 (- 8 �E��� : ;=<F>0@A 2 �G�������.�DC1��H (1)
Invokingaxisymmetry, thestresstensorhastheformI 4KJLNMPO�O � MPO >� MRQ�Q �MPO > � M >0>ST H
Consideran infinitesimalsliceof materialof height U � . Theforcesactingon sucha sliceareaveragedandsummed.Theaverageverticalstressis clearlygivenby
3M >0> . We denoteby V andW thebulk densityandgaspressurerespectively, andby
3V and3W , their respective
averagevalues.ThevariousforcesareX weightof solid: Y 3V )Z- 8 ���.� U � ;X if3[]\ is the averagewall shearstressand
3M \ the averagenormalstresson the wall,thereareupwardforcesof
(D)Z-"���.� U � 3[ \ and(D)^-"�E���_-9`E���.� U � 3M \ ;X upwardpressuredueto thewall:
(+)9-"�E���_-9`E���.� U � 3W �E�P� � � ;X pressureatbottom:3W �E�P� � �.)Z- 8 ���.� , andtop: Y � 3W �E�P� � �ba U0W �.)Z- 8 ���ca U ��� ; thiscreates
a force Y )Z-��E��� 8 U0W Y (D)Z-"���.�_-9`��E��� 3W ���d� � � U � ;X averageverticalstressat bottom:3M >0> �E��� , andtop: Y � 3M >0> �E���=a U M >0> � ; (compressive
stressesaretakenaspositive for granularmaterial);this createsa force Y )Z- 8 �E��� U M >0> Y(D)9-"���.�_-9`E�E��� 3M >0> U � .Theresultingbalanceof forcesequationise > 3M >0> a e > 3W a (+-9`��E���-"���.� 3M >0> Y (-"���.� � 3[ \ a 3M \ - ` ���.� �fa 3V 4g��HOnthewall, thelaw of sliding friction applies3[]\ 4ih \ 3M \ � (2)
whereh \ is the coefficient of wall friction. The averagestresstensor
3Iis diagonal;
in other words, both3M O�O and
3M >0> are principal while3M O > 4j� . Notice that
3M \ ���.�k4
4 K.A. KRISTY AND P.A. GREMAUDl�m�n ocpdq�r.sutPv�vwqEr�syx�n�zF{ o pPq�r.sutd|0|�qEr�s where}�~ {�pPq�r.s����9�EqEr�s . Theratio of averageverticalstress�t |0| to averagewall stress�tP� � � �td��td|�|=� (3)
is takenasdependingonthegeometryonly, see� 4 for moredetails.Theaboveassumptionis the pendantto Janssen’s analysis,[6], [8], [10], which is routinely usedin verticalbunkers. In (3),
�dependson the geometryof the containerandhasto be determined
throughexperiments[8], [13] and � 4. Equations(2) and(3) thenyield� | �t |0| x � | �� x�� �9�EqEr�s�"q�r.s �t |0|�� ��"q�r.s qG�7��x�� � q�r.s s � �t |0| x �� �g��� (4)
Let � bethedensityof thesolid particles,which is assumedconstant.Thegasdensity,denotedby � , is anunknownfunctionof timeandposition.Thesetwo quantitiesarelinkedthroughthebulk density� � ���*� � x�qu� � �*�%s � � (5)
where �*� is thevolumefractionoccupiedby thesolid. Generally� is at leastthreeordersof magnitudelargerthat � , andthus �*��� �� � (6)
The averagebulk densityis consideredasa functionof the averagemajor consolidatingstress,here �tP|0| , i.e., �� � �� q �td|�|�s . Varioussuchrelationshave beenproposed,see[10],� 6.2,for areview. Thosemodelstypically makesensefor only a limited rangeof valuesof�t |0| . Following [7], weassume �� � �.� qu�Nx �t |0|t � s�� � (7)
where�� ¢¡¤£¥� is thecoefficientof compressibilityof thematerial,� �§¦ � and t �¨¦ �arematerialconstants.Wereferto [3] and[2] for respectively theoreticalandexperimentalinvestigationsof theprecisenatureof thebulk density/stressrelation.
Assumingpowder is neitherenteringnor leaving the system,the total mass© of thesolid is conserved,leadingtoª�«�¬®G¯° � o qEr�s ���± �t5qEr ��² s ³c´�r,� © µ·¶¹¸© � ²»º �P� (8)
Applying thecontinuityequationto bothgasandsolidphases,we get� � x&¼¾½wq �^¿yÀ s$�Á� (9)� 5 qu� � �� s �.à x¢¼¾½  qu� � �� s � ¿fÄ Ã �Á� � (10)
where¿fÀ and ¿fÄ arethevelocitiesof thesolid andthegas,respectively.
POWDERCONSOLIDATION 5
Usingaxial symmetryandapplyingtheaveragingoperator(1) to (9) yieldsÅ�Æ]ÇÈ"É ÊË"Ì�Í.Î È Ì�Ë"Ì�Í.Î%Ï�Í�Î.ÐcÑ�Ò Ó�ÌEË�ÌEÍ�Î�Ï�Í.Î É ÊË"ÌEÍ�Î Ô�Õ�Ö=×FØ0ÙÚ Å ØDÛ È Ì�Ü�Ï�Í.Î.ÐcÑ�Ò Ø Ì�Ü�Ï�Í.ÎÞÝßÜDàwÜâáiãPäFurther, assumingthegrainsin contactwith thewall to move tangentiallywith respecttoit, we observe
ÐcÑ�Ò Ó*Ì�Ë"ÌEÍ�Î�Ï�Í.Î5á�Ë9åæÌ�Í.Î.ÐcÑ�Ò Ø ÌEË�ÌEÍ�Î�Ï�Í.Î . Elementarymanipulationsthenleadsto Å Æ ÇÈ'É çË9ÔwÌEÍ�Î Å Øéè Ë Ô Ì�Í.Î È Ð Ñ�Ò Øëê áiã�äNeglectingfluctuationsin theradialdirectiongivesÈ ÐìÑ�Ò Ø�í È ÐcÑ�Ò Ø . Similarprinciplescanbeappliedto local conservationof gas.Thoseconservationlaws thenreadÅ1ÆbÇÈ'É çË Ô ÌEÍ�Î Å Ø è Ë Ô ÌEÍ�Î ÇÈ ÇÐìÑ0Ò Ø ê áÁã�Ï (11)Å�Æ5î Ì ç�ï ÇÈð Î Çñ.ò'É çË Ô ÌEÍ�Î Å Ø î Ë Ô Ì�Í.Î Çñ Ì çóï ÇÈð Î ÇÐRôbÒ Ø ò áÁã�ä (12)
In additionto (3) and(7), two additionalconstitutive equationsareconsidered.First, thegasis assumedto beidealandisothermalõ ñ á õ Úñ Ú Ï (13)
where
õ Ú and ñ Ú areconstantreferencevalues.Second,pressuregradientandvelocitiesarerelatedthroughDarcy’s law, which readshereÇÐdôëÒ Ø ï ÇÐìÑ0Ò Ø á ïóö Ì ÇÈ Î Å Ø Çõ Ï (14)
whereö is thepermeability, takenasa functionof theaveragebulk density. We take [8]ö Ì ÇÈ Î5á ö Ú Û ÇÈÈ Ú ÝZ÷cø Ï (15)
whereö Ú and È Ú arereferencevaluesand ù is apositiveconstant.Theparametersú , ûPü ,È ü , ù , ö Ú and È Ú appearingin (7) and(15)haveto bedeterminedexperimentally.Wenow eliminatevelocitiesfrom thesystem.Puttingtogether(14)and(11)yields,after
integrationbetween0 andagenericpointÍÕ&ØÚ Ë Ô Ì]ýÍ�Î Å Æ ÇÈ Ì�ýÍ1Î�àfýÍ É Ë Ô Ì�Í.Î ÇÈ Ì�Í.Î Û ÇÐ ôëÒ Ø ÌEÍ�Î É ö Ì ÇÈ ÌEÍ�Î�Î Å Ø Çõ Ì�Í.ÎÞÝï Ë Ô Ì�ã�Î ÇÈ Ì�ã1Î Û ÇÐRôbÒ Ø Ì�ã�Î É ö Ì ÇÈ Ì�ã�Î Î Å Ø Çõ ÌEã1ÎÞÝ�á�ã�ä
Assumingno gasleavesthe (closed)containeratÍþáÿã
, naturalboundaryconditionsatthebottomarethen
ÇÐ ôëÒ Ø Ì�ãPÏ�� Î5á�ã andÅ Ø Çõ Ì�ã�Ï�� Î5áiã . Thepreviousequationcanbesolved
forÇÐdôëÒ Ø Ì�Í.Î , leadingtoÇÐRôbÒ Ø Ì�Í.Î5á ï çÇÈ Ì�Í.Î_Ë9ÔwÌEÍ�Î Õ&ØÚ Ë Ô Ì]ýÍ1Î Å�ÆbÇÈ Ì�ýÍ1Î�àfýÍ ï¤ö Ì ÇÈ ÌEÍ�Î�Î Å Ø Çõ Ì�Í.Î%ä
6 K.A. KRISTY AND P.A. GREMAUD
Using(13)andtheaboveexpressionfor ������ � ��� , (12)canberewritten in termsof pressure� eliminating � from theproblem.The full systemof equationscan now be considered. The three unknowns of the
problemarethe averagedgaspressure�� , the averagedvertical stress �� � andthe heightof the powder � ���� . We obtain the following systemof integrodifferentialequationsin� ������������� �����!�"�#� � �����$�&%')(+* �,- �/.�0 ��#* �� �, .�0 �,1* (243 ��5� .576 ��8 (�, * (- �)9;: < 2 3 ����.�0 �, (16)*8)(=* �,- �/.5� ��?>;.5 ����A@ > ��- .� �,B.5 ��A��&%C. �� � @D. ��E@ F 24G ���2 ��� �� � * F2 ��5� IHKJL@ 2 G ��5���NM �� � @ �,O� (17)PQ %R:DS?T 0IU< 2 3 ��� �,WV �� �� P��������XAY P��Z (18)
Thesystemis completedby initial andboundaryconditions��A)[\����� %]� < ��� � � ����������^%_���`. ��A �N����� %a�N� ��b � ����������c%d��e 0If �where � e 0If is the value of atmosphericpressure. The boundarycondition at �g%h�
correspondsto asolid bottomedcontainer. Thiscanbeeasilymodifiedto includethecasewhensomematerialexits throughanoutletat thebottom. A mathematicalanalysisof theaboveproblemin thecaseof a cylindrical container, i.e.,
24G ��5�i%_� , canbefoundin [4].A usefultool for the presentkind of equationsconsistsin mappingthe entireproblem
into a fixedspatialdomain. Upon discretization,this type of formulationleadsto betterstability propertiesthanformulationswherethegrid moveswith thematerial,suchas[8].Let j %_�/k � ���� �b j �����i% ��O��l����� � j �����i% �� � ��������ZBasedon theserelations,theprevioussystemof equations(16), (17)and(18)yields�&%m)(=* ,- �n6N. 0 �#* j .5o�� � G ����� I��� 9p* �, 65, G . 0 � * j , G .5o � � G ����� ���� 9 (19)* (243 � I��� j � .�o 6 �� (, * (- � 9;: o< 2 3 � ���� j � 6 , G . 0 � * j , G .5o � � G I���� I��� 9* (� 3 ���� �(+* ,- �/.5o/q�?>;.5o��r�K@ >1�- � 3 ���� , G .5o � .5o��A��&%a.5o � @s.5o���@sF 24G � I��� j � � I���2 � ���� j � � * F � I���2 � I��� j � 6 HKJn@ 2 G ��� 9 M � @ � ����t,u� (20)PQ % � I���l: v< 2 3 � ���� j ��,=V � j ������XAY j � (21)
wherewe havedroppedthebarnotation.Initial andboundaryconditionsbecome�O j �����s%w� < j ��� j1x �N�y(z��� (22)� �({�����i%_���R.5o��b��N����� %C�N�|�b)(������}%~�re 0If � (23)
POWDERCONSOLIDATION 7
Thechangefrom �!�����K�p�y��� toageneralfunction �!� ��� is farfrombeingbenign. If �!� ���u��B�#�����y� , (21) canbe solved for �s�I��� : �s�I����� ������E���^��D�=��� �����������O�{�/�O� � . Taking the
derivative,onecanalsoobtainanexplicit expressionfor ��� �I��� . Thoseexpressionscanbethensubstitutedin (19) and(20) to fully eliminate � from the system[4]. For generalfunctions � however, (21) is a nonlinearequationfor �D����� , which cannotbeexpectedtoalwaysyield explicit solutions. To maintaingenerality, �D����� is not eliminatedfrom thesystembeforediscretizationbut is rathercomputedasan unknown alongwith stressandpressure.
3. DISCRETIZATION
Theseequationsarediscretizedandsolvedin thefixedrectangulardomain� ¡N��¢��5£���¡���¤B� .Set ¥��!�;¢z¦¨§ andlet ��©A�;��ª�«s¢���¥�� ªu�;¢{��¬y¬y¬y��§®p¢ . Thesemidiscretizedvariablesarepressure#�I���+�±° ¯ � ��������¬y¬y¬���¯b²E�I���t³ andstressµ�����+�¶°·´ � �I���¨�y¬�¬y¬y��´¸²������t³ . Note thattheremainingunknown, � , dependsonly on time. Theboundaryconditionsat �!�;¢ read¯b²¸¹ � �º¯u»�¼I½ and ´¸²W¹ � �º¡ . A secondorderspatialdiscretizationfor bothvariablesisused.Thediscretizedoperatorsare¾i¿
: §h£n§ matrix correspondingto a secondordercentereddiscretizationof À5Á withpressureboundaryconditions(at �#�p¡ and �!�¢ ),Ã
: §h£n§ matrix correspondingto a secondordercentereddiscretizationof À Á¨Á withpressureboundaryconditions(at �#�p¡ and �!�¢ ),¾¸Ä
: §Å£n§ matrix correspondingto a secondorderbackwarddifferentiationformula(BDF [1], [5]) for À Á with stressboundarycondition(at �#�¢ ).Although moresophisticateddiscretizationmethods,suchasBDF basedalgorithms[1],shouldbeconsideredin spaceaswell in general,thediffusionlikedynamicsof thepresentproblemaresufficiently simpleto mitigatethis point of view. Theconstructionof
¾i¿andÃ
is elementary. For¾¸Ä
wetake
¾¸Ä � ¢Æ ¥7ÇÈÉÉÉÉÉÉÉÉÉÊ«+Ë Ì±«�¢ ¡Í¬�¬y¬s¬�¬y¬ ¡¡¶«+Ë Ì±«�¢ ¡Í¬�¬y¬ ¡¬y¬y¬ ¬y¬�¬s¬y¬�¬D¬y¬�¬s¬�¬y¬s¬�¬y¬ ¬y¬y¬¡Å¬y¬�¬ ¡¶«+Ë ÌΫ�¢ ¡¡Å¬y¬�¬s¬y¬�¬ ¡¶«+Ë Ì±«�¢¡Å¬y¬�¬s¬y¬�¬D¬y¬�¬ ¡±«+Ë Ì¡Å¬y¬�¬s¬y¬�¬D¬y¬�¬s¬�¬y¬ ¡¶« Æ
ÏÑÐÐÐÐÐÐÐÐÐÒ ¬In addition,thefollowing discretizedintegraloperatorsareintroducedÓBÔ � ÕÂ�^� ¢ Ö× ²W¹ �ØÙ�Ú �cÛ Ù Õ Ù and � Ü Ô � ÕÂ���)©b� ©ØÙ�Ú �lÛ Ù Õ Ù �whereÛ denotestheweightsof thequadrature.Threeadditionalvariablescorrespondingtothederivativesof thethreemainthreeunknowns ¯ , ´ and� arealsointroduced,ÝÞ�_� ¼ ¯ ,ß �à�{¼�´ and ás�à��� . Thespatiallydiscretizedproblemis��¼)¯â�àÝ (24)
8 K.A. KRISTY AND P.A. GREMAUDã äÂå)æ?çÞèé?ê+ë�ì çîíïàðDñ}ò{ó4ô ç ó è ë è�õ÷ö ç è�õ íïøðsñ¸ù�úcô (25)ç æûµüLë å æè ç æé ê�ñ ò ó ç ó è õè ü ñ ù ú ô;ë�ýBþ å û ü è õ ö ê çÿíïpýµþ å û ü è õ ðEñ ù úWê ôç æï�ü å)æ=ç;èé ê�� å ó���úWê)ó�� ó��éAï�ü è õ ñ ù úWñ�}ó���� æ����uå ó���ì�� ö ��ú�� ï � í êã ä ñ¸ù�ú��sñiò{ó�� � ï�û õû å ï ð�ê ú����� �Bç � ïû å ï ð�ê å���� � û õ ê�� ï è (26)��#å ó���ì�� ö ��ú�� ï � í ê�! ú ä ö (27)� tï ä í (28)ã ä ï#" þ�$ û ü å ï ð�ê è�% ��&Aå ó���ì�� ö ��ú�� ï � í ê�� (29)
where è å úWê and è õ å úWê arevectorsand ð ä(' )+* �-,.,-,�� )!/�0 . The two vectors �� æ and �� �resultfrom thepresenceof boundaryconditions.Also, vectorto vectormultiplication intheaboveexpressionsis takento becomponentwise.Theproblemhastheform�1 ó ä ì��ã&ä2�uå ó���ì�� ö ��ú�� ï � í ê��ã&ä43�å ó�� ã � ã ��ú�� ï � ã ê���1 ú ä ö� ï ä íã&ä2&Oå�ã � ã � ã ��ú�� ï � ã êThis correspondsto ansemi-explicit index 2 DAE or equivalentlya fully implicit index 1DAE [1]. A linearly Implicit Eulertimediscretization[5], [11] of (24)-(29)leads56666667
8 9;:�<!8 = = = =9;:�<+>@?A 9;:�<1>@?B 9;:�<1>@?C 9;:�<1>@?D 9;:�<+>@?E 9;:�<1>@?F9;:G<!H!?A 9;:�<IH!?B 9;:�<JH!?C 9;:�<JH!?D 9;:G<KH!?E 9;:�<KH!?F= = 9;:G<18 8 = == = = = L 9;:�<9;:�<!M ? A 9;:G<!M ?B 9;:�<!M ?C 9;:�<!M ?D 9;:�<!M ?E 9;:�<1M ?FNPOOOOOOQ56666667R ?TS A 9UR ?V?TS A 9#V?WG?TS A 9XWG?YZ?TS A 9XYZ?[\?TS A 9U[\?] ?TS A 9U] ?
NPOOOOOOQ ä_^a`56666667V ?>@?H!?WG?]�?M ?NPOOOOOOQ
wherethesubscriptsdenotederivativeswith respectto thecorrespondingvariables.Thefirst partialderivativesof
�and
3areevaluatedthroughfinite differences.
4. COMPUTATIONAL RESULTS
Two caseswith differentgeometriesareconsidered.Thefirst correspondsto a cylinderandthesecondto abunkerconsistingof acylinder for theupperportionandaconefor thelowerportion. Thedatais initializedasfollows.
Thevaluesof variousparametersin themodelhavebeentakenasto representarealisticsituation bdc ä , �!e é äf� ã{ã
lbs/ftgè c ä_h{ã lbs/ftg �\i ä;æ�ãkjml ftllbsj *
sj *n c ä;æ-o lbs/ft
ü p ä_qè i ä�r{ã lbs/ftg � � äsut!vAåxwzyI{ ê
POWDERCONSOLIDATION 9
Theatmosphericpressureis |d}�~x�f���k�!���k� � lbs/ft� .Thedeterminationof theratio � betweenwall stressandverticalstress,see(3), is more
delicate. In the caseof a perfectlyplasticflow, sucha coefficient couldbe derivedfroma plasticity model,Mohr-Coulombfor instance[10]. Although this is oftendoneaspartof ananalysisa la Janssen,suchanapproachis not consistentwith themechanicalstatesconsideredheresinceonecannotassumethematerialto have reachedyield. Thevalueoftheratio � does,however, dependon thegeometry. For plasticgranularflows, thestressesareusuallybelieved to be closeto an active statein a vertical cylindrical hopper, whilethey tend to be in a passive statein converging conical hoppersfor instance[10], [15].Experimentalevidence[13] seemsto point to the fact that,evenhere,a similar behaviorcanbe observed. More precisely, the valueof � tendsto be muchlower in a convergingconicalsectionthanin averticalcylindrical one.Thevaluesbelow aretakento reflectthis.
Let us considerthe cylinder with radius ������� � ft (andthus ������ ) with an initialheightof powder �����1����� ft. The ratio � is takenas �X���J�J� . An initial pressurefield,|�� , consistentwith theboundaryconditionsin (23) is givenas
| � ���@����| }�~x��� �; �����+��-�!� ����¡¢� ������+� � � �¤£¥�In thecalculationsbelow, ¦§�f�I�!� and ¦X¨_��¨G�I©aª«���I�1� .
For thesecondcase,theradiusof thecylinder is ���¬��� � ft. The radiusat thebottomof thebunker is ��� ft, andtheheightof theconicalpart is � ft. Thus, �®�¯�°���±� . In orderto have approximatelythesamemassof materialin this hopperasthefirst case,thetotalinitial heightof thepowderis �����1���¬�²� +�!³ ft. Theconsideredvaluesof � are1/3 in thecylindrical part,and �´�µ� in theconicalpart. Thesameinitial pressurefield is taken. Inaddition, ¦¶�f�I�1� and ¦X¨_�_¨��!©·ª«���!�!� asbefore.
Figure2 givesatypicalillustrationof thebehavior of theproblemin acylinder. NotethatFigure2 is relatedto theoriginal geometryof theproblem,i.e., thevariableis � not ¸ . Asexpected,asettlementof thepowderis observed.For bothgraphs,thetopline correspondsto thepositionof thetopof thepowdercolumn,i.e.,thefreeboundary. Further, thepressurefield is foundtoasymptoticallyconvergetoauniformpressurevaluecorrespondingto |K}¹~x� ,the atmosphericpressure,i.e., equilibrium of the pressureis established.The stressfieldis alsofound to converge to a stationarydistribution ºm» which, in termsof the originalvariable� , is solutionto¼@½ ºm»¾¡ ��¿���@�TÀ�Á �kºm»¬ �«��ºd»\�¢�Ã��Ä �¿Å��¿Å¢�\»Æĺm»U���\»Ç���_��Äwhere �\» is the asymptoticvalueof the heightof the powder column. Figure4, left,
illustratestheconvergenceto theasymptoticstationarystateafter2000s.Figure3 givesatypical illustrationof thebehavior of theproblemin abunkerconsisting
of acylinderastheupperpartandaconein thelowerpart. Againsettlementof thepowderis observedby consideringthetop line of thegraphs.As in thepreviouscase,thepressurefield asymptoticallyconvergesto a uniform pressurevaluecorrespondingto |d}¹~x� . Thestressfield alsoconvergesto a stationarydistribution ºm» which, in termsof the original
10 K.A. KRISTY AND P.A. GREMAUD
2120
2130
2140
2150
2160
2170
2180
2190
2200
2210
2220
0 500 1000 15000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
pressure (lbf/ft2)
time (s)
heig
ht (
ft)
20
40
60
80
100
120
140
160
180
0 500 1000 15000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
stress (lbf/ft2)
time (s)
heig
ht (
ft)
FIG. 2. Calculatedpressureandstressfieldsin thecylindrical geometry;È�ÉUÊ�Ë�Ë , ÈGÌÆÉUÍ�Ë�Ë , ÌÆÉÏÎ�Í�Ë�Ë .
2120
2130
2140
2150
2160
2170
2180
2190
2200
2210
2220
0 500 1000 15000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
pressure (lbf/ft2)
time (s)
heig
ht (
ft)
20
40
60
80
100
120
140
160
180
0 500 1000 15000
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
stress (lbf/ft2)
time (s)
heig
ht (
ft)
FIG. 3. Calculatedpressureandstressfields in thecylinder-on-a-conegeometry;È�É#Ê�Ë�Ë , ÈGÌÏÉ#Í�Ë�Ë ,Ì´ÉÐÎ�Í�Ë�Ë s.
POWDERCONSOLIDATION 11
0 1 2 3 4 50
50
100
150
200
250
height(ft)
stre
ss(lb
/ft2 )
0 1 2 3 4 50
50
100
150
200
250
height(ft)
stre
ss(lb
/ft2 )
FIG. 4. Calculatedstressfield in theoriginalgeometry;left: cylinder, right:cylinder/conegeometry.
variableÑ , is solutiontoÒ@Ó-ÔdÕ¬Ö�×�Ø�ÙØÛÚ Ñ+Ü ÔmÕ¾Ý ×ØÛÚ Ñ+Ü ÚxÞ�ß Ö Ø Ù Ükà ÔmÕ¬Ö¥á Ú ÔmÕ Ü¢âÃãkä ã¿å�Ñ¿å�æ Õ äÔ Õ Ú æ Õ Üçâ_ãkäwhereæ Õ is theasymptoticvalueof theheightof thepowdercolumn.Figure4 illustratestheconvergenceto theasymptoticstationarystateafter2000s.
To verify the resultsof the calculations,an experimentmust be designedso that theparametersfor themodelcanbedetermined.Veryaccuratemeasurementsof theseparam-etersareneededto verify thecalculatedresultswith experimentalones.ConsiderFigure5which shows thehalf-life of theover-pressureat thebottomof a cylindrical bunkerversusthecompressibilityparameterè . Thehalf-livesrangefrom nearlyzeroto approximately650for è rangingfrom zeroto four. It canbeshown thatthehalf-life of theover-pressureis alsosensitive to otherparametersin themodel[12].
5. CONCLUSIONS
Existingmodelsfor powderconsolidationhave beenextendedto generalaxisymmetricdomainsin orderto take into accountgeometricaleffects.In thisprocessandaswith thoseprior models,averagingonhorizontalcrosssectionsplaysa fundamentalsimplifying role.The resultingsystemconsistsof anessentiallyparabolicPDE,andan ODE in spaceandanintegral equation.In spiteof thestrongnonlinearcoupling,thosecanbethoughtof asrespectively equationsfor thegaspressure,theverticalstressandtheheightof thepowder.A fundamentalandproblematicmaterialparameteris theratio à betweenwall stressandverticalstress.
Unfortunately, someof theparametersenteringthemodel,suchas à or thecompressibil-ity è , arenot easyto measurein a reliableway. To make mattersworse,thepresentwork
12 K.A. KRISTY AND P.A. GREMAUD
0 0.5 1 1.5 2 2.5 3 3.5 40
100
200
300
400
500
600
700
pres
sure
hal
f−lif
e
β
FIG. 5. Calculatedhalf-life of over-pressureversustheparameteré .
illustratesthatevenrelatively small variationsin thosecoefficientscanhave a noticeableeffectonthesolution,see,e.g.,Figure5. Thismaypartiallyexplain therelativesparsityofexperimentalresultsfor thepresenttypeof problems.
Themaincontributionsof thepaperarefirst acarefulgeneralizationof earliermodelsinorderto take geometricaleffectsinto accountandsecondthedesignandimplementationof a robustandefficient numericalalgorithmfor thecorrespondingequations.While theyextendearlier results[4], [8], [12], the presentedcomputationalexperimentsseemto beconsistentwith them.
Variousaspectsof thepresentmodelarequestionable,primarily theuseof apseudoone-dimensionalformulation throughspatialaveraging. Although a truly multidimensionalapproachappearsto be desirable,its precisenature,especiallyin term of constitutiveassumptions,is unclear.
ACKNOWLEDGMENTS
Theauthorsarealsogratefulto SteveCampbell,Tim Kelley andTony Royal for helpfuldiscussions.
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POWDERCONSOLIDATION 13
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