Compaction of the diamond Ti 3 SiC 2 graded material by the high speed centrifugal compaction process

Transcription

1 Archves of Materals Scence and Engneerng Volume 8 Issue November 7 Pages Internatonal Scentfc Journal publshed monthly as the organ of the Commttee of Materals Scence of the Polsh Academy of Scences Compacton of the damond T SC graded materal by the hgh speed centrfugal compacton process V. Mtyushev a, L. Jaworska b,c, *, M. Rozmus c, B. Królcka b a Department of Mathematcs, Pedagogcal Academy, ul. Podchor¹ ych, -84 Kraków, Poland b Department of Technques, Pedagogcal Academy, ul. Podchor¹ ych, -84 Kraków, Poland c Insttute of Advanced Manufacturng Technology, ul. Wroc³awska 7A, - Kraków, Poland * Correspondng author: E-mal address: lucyna.jaworska@os.krakow.pl ceved 5..7; publshed n revsed form..7 ABSTRACT Purpose: Sedmentaton of partcles n a vscous flud s a man physcal problem n flud mechancs. Sedmentaton s benchmark of one of the technca; methods to produce the functonally graded materals (FGM wth a contnuous spatal change of mechancal propertes. The am of the research was executon of mathematcal calculatons of the phases dstrbuton for the phase graded damond-t SC compacts whch were verfed wth phases dstrbuton n compacts after the hgh pressure hgh temperature snterng process. Desgn/methodology/approach: In ths paper, we construct a mathematcal model of FGM basng on the modfcatons of the Stokes formula. We proposed an algorthm to descrbe sedmentaton of the group of sphercal partcles of dfferent szes and dfferent materals. Man calculatons for ths system and real condtons of the hghspeed centrfugal compacton process are made usng the Barnea-Mzrah equaton. Deposton process was carred out usng the ultra centrfuge UP 65M wth rotatonal speed of 5 to 5 rpm. Partcle sze dstrbuton for the damond and T SC powders were measured usng Shmadzu apparatus. Fndngs: The results of calculatons and mcroscopc analyss are compared. The obtaned results of mathematcal calculatons demonstrate that for the consdered damond-t SC suspensons the obtaned compact has the structure of the lamnate what confrmed mcroscopc analyss. Practcal mplcatons: The mathematcal smulatons usng our algorthm show that t s possble to obtan contnuous concentratons of the both materals wth approprate ntal suspensons. Thus our method allows to obtan graded materals. Orgnalty/value: Ths mathematcal model gves possblty of use to descrbe sedmentaton of the group of sphercal partcles dfferent materals and dfferent szes. Keywords: Hgh-speed centrfugal compacton process; Sedmentaton; Barnea-Mzrah equaton MATERIALS MANUFACTURING AND PROCESSING Copyrght by Internatonal OCSCO World Press. All rghts reserved

2 V. Mtyushev, L. Jaworska, M. Rozmus, B. Królcka. Introducton I n Sedmentaton of partcles n a vscous flud s a classcal physcal problem n flud mechancs. It s the benchmark of one of the techncal methods to produce lamnates, n partcular, the functonally graded materals (FGM wth a contnuous spatal change of mechancal propertes [,,,4,5]. Such materals have advantages on the classc lamnates and other compostes, because the nterface between phases yelds jumps n the thermal expanson and the elastc modulus of meda. In the same tme, FGM create a slowly changng gradent of the thermoelastc propertes n meda. Hence, the thermal flux and the stress tensor change also slowly n such materals. A sngle heavy sphercal partcle falls n a vscous flud under the gravtatonal feld accordng to the Stokes settlng velocty [6] U s d ( c g, ( 8 where d denotes the dameter of the sphere, and the c denstes of the partcles and of the flud, the dynamc vscosty of the flud, g the gravtatonal acceleraton. Formula ( s vald for lamnar flow when the ynolds number U d s satsfes the nequalty 4 [6]. The latter restrcton holds for slowly fallng partcles. We now brefly present recent theoretcal and expermental works devote to dfferent aspects of sedmentaton. R. B. Jones & R. Kutteh [7] appled the Stokesan smulatons to compare the hydrodynamc nteractons between collodal partcles n unbounded flud and near a hard wall. The results showed that partcles n unbounded flud there s a hgh degree of symmetry n the confguraton of the partcles whch s broken by the presence of the walls. A smlar method based on statstcal mechancs were used n [8] to study random clouds of the partcles. The gravtatonal settlng of small partcles were numercally nvestgated n [9]. The effect of sde walls on the velocty fluctuaton were nvestgated theoretcally n [,] and expermentally n [,]. A model of the velocty fluctuaton were presented n [4]. Ths model showed that there s a crtcal stratfcaton above whch velocty fluctuatons changed sgnfcantly. Lattce-Boltzmann smulaton were appled to monodsperse and polydsperse suspensons [5]. It was verfed that partcle velocty fluctuatons were suppressed by no-slp condton on the walls. Kynch s cnematc sedmentaton model appled to hgh concentraton suspensons and ts generalzatons are descrbed n [6]. One can also fnd a revew of recent expermental and theoretcal works n [7]. It follows from the above revew that the sgnfcant success has been reached n nvestgatons of the gravtatonal sedmentaton whch s characterzed by the constant gravtatonal forces. The queston of the centrfugal sedmentaton when the external forces actng on suspenson essentally exceed the gravty and change n the tme s not deeply nvestgated theoretcally and expermentally. In the present paper, we construct a mathematcal model of the centrfugal sedmentaton of dlute suspensons, apply ths model to damond and T SC partcles and compare the theoretcal results to the experments. Takng nto account hydrodynamc nteractons of partcles and the effect of walls t s possble to correct formula (. One can fnd varous modfcatons n [6] and papers cted theren. In the present paper, we use the Barnea-Mzrah correcton ( U U ( s / 5 ( exp ( Where denotes concentraton We have chosen formula (, snce t gves more realstc results than other correctons [6] n our experments presented below. It s worth notng that sedmentaton for hgher concentratons are descrbed by non-lnear parabolc equatons. Dlutent suspensons when s small are suffcently well descrbed by equaton (. In the present paper, we consder the case =., hence formula ( can be assumed as the benchmark of our further calculatons. The hgh-speed centrfugal settlng requres verfcaton of the theoretcal results based on formulas (-(, snce the gravtatonal acceleraton g s replaced by the centrfugal acceleraton g whch exceeds g at least fve tmes. Moreover, the centrfugal acceleraton depends on tme t. We dscuss these theoretcal questons n Sec. and. In Sec. 4 new formulas for the hgh-speed centrfugal settlng are deduced and appled to Hgh Speed Centrfugal Compacton Process.. Descrpton of the mathematcal model.. Geometry of vessel.. Geometry of vessel The geometrcal scheme of the volume s presented n Fg.. It s a tube vessel wth the crcular base. The dstance between the centre of rotaton and the top of the vessel s denoted byr mn and fxed equal to 5.5 cm. The dstance to the bottom r s equal to.88 cm. So, the heght of the vessel s 7.8 cm. The axs of rotaton s perpendcular to the axs of the tube. Sphercal partcles are unformly dstrbuted n the vessel, hence we have at the ntal tme t a unform suspenson. At the begnnng of the experment the gravtatonal forces act to the partcles but not essentally change ther poston. In the process of the accelerated rotaton the centrfugal forces exceed the gravtatonal forces more than fve tmes. Ths mples that the partcles fall from the top to the bottom of the vessel and the gravtatonal effects can be neglected n such experments Archves of Materals Scence and Engneerng

3 Compacton of the damond T SC graded materal by the hgh speed centrfugal compacton process Equatons (-( become centre of rotaton d ( c ( U ( r( (, (4 8 / 5 ( exp ( r mn r Fg.. The geometrcal scheme of the rotor.. Non-statonary sedmentaton In order to apply formulas (-( to the hgh-speed centrfugal sedmentaton we have to make modfcatons. st, the constant g s replaced by the functon g( r( (, where r ( s the dstance whch a partcle passed from the top rmn 5.5cm of the vessel to the bottom at the tme t; ( s the angular velocty. A typcal plot of ( s gven n Fg.. 5 [rad/s] t [s] Fg.. Dependence of the angular velocty on the tme t. Tme of acceleraton t = 47 s, tme of constant angular velocty t = 5 s, tme of brakng t = 8 s, constant angular velocty rpm The functon ( s pecewse lnear and has the form at, tt, (, ttt, ( atb, t tt, where t s the tme of acceleraton, t t the tme of constant angular velocty, t t the tme of brakng, t the total tme of the process; a, a and b are approprate constants. where U ( denote the velocty of the partcle at the tme t. dr Usng the relaton ( U ( we arrve at the lnear dt dfferental equaton of frst order dr d ( ( ( c r( (, dt 8 / 5 (5 ( exp ( wth the ntal condton r( r. Applyng the method of mn separated varables we obtan r( r where t mn expd ( ( d. ( c 8 ( / ( (, 5 exp ( (6 The latter ntegral s easly calculated for the functon ( gven by ( a t, tt, a (7 ( t t, ttt, a a t t t abt bt, t tt Usng formulas (-(7 we can determne the path r ( of any partcle (see example n Fg. 4. Formula (6 s vald up to t t when the functon r( attaches the mal value r ( r. 88 (8 correspondng to the bottom of the vessel. The relaton (8 can be consdered as a number equaton wth respect to t. Ths equaton s easly solved by the method of successve approxmatons, dr snce the dervatve ( U ( s a postve functon (see dt (4, hence r ( s an ncreasng functon. Thus, equatons (6-(8 produce an algorthm to compute the settlng tme t of the partcle of the dameter d. Consderng t as a functon Volume 8 Issue November 7 679

4 V. Mtyushev, L. Jaworska, M. Rozmus, B. Królcka t ( t d we can estmate the sedmentaton regme of the partcles of dfferent szes. Example s presented n Fg.. t [s] d [-6 m] Fg.. Dependence of the settlng tme t on d for T SC One see at Fg. nstablty of the dependence t t( d 6 near the pont d. m,.e., small perturbatons of the dameter d produce large devatons of t. Ths fact can be essental for small partcles, because the dstrbuton of the szes s tabulated wth an error whch can drastcally change the tme of settlng. Here, the dffuson processes whch can take place for small partcles submcrometrc or nanometrc [7] are not taken nto account. Consder many sphercal partcles of dfferent szes made from the same materal. Usng equatons (6-(8 one can observe the detals of the polydsperse settlng and ts dependence on the sze of the partcles. An example of the T SC partcles s presented n Fg. 4. One can see that not all partcles of the dameter d =.5 reach the bottom durng the process. r The rato r( t corresponds to the part of not settled rrmn partcles to the end tme t. The rato settled and non-settled partcles s equal to the rato of the vertcal segments cut by the curve r ( for d =.5 at the tme t. It s convenent to use the dscrete densty dstrbuton of the szes d, n,,,..., n, where d denotes the dameter of the-th partcle, n the rato of the number of partcles wth the dameter d to the total number of partcles. M Introduce a value M proportonal to the total mass of partcles n d. (9 The part of the mass of the settled partcles W ( at the tme t s calculated by formula W ( d, ( M rr where the sum n ( s taken over all r ( equal to r. Here r ( s calculated by (6-(7 wth d d. An example of the functon W ( s presented n Fg.5. Here, the computatons are performed for the damond partcles. The dstrbuton of the partcles d, n were measured by Shmadzu s apparatus. It s presented at Fg. 6. r [cm] 8 6 dameter d [ -6 m] t [s] Fg. 4. Path r( calculated by (6-(8 for varous d. gme of rotaton s descrbed n Fg.. Two vertcal lnes near t = corresponds to the change of the angular velocty W t [s] Fg.5. Dependence of the concentraton W on t for damond powder for the regme of rotaton descrbed n Fg. The proposed method can be developed to partcles made from dfferent materals. Let us take two dfferent materals: damond (.55g / cm and T SC ( 4.5g / cm. Let the total masses of damond and of T SC are equal. Introduce the parts of the total mass of all settled partcles ( and W T ( for damond and T SC, respectvely, by formula ( wth the coeffcents.5. Then Wd ( W (, ( W ( W ( d T W d Archves of Materals Scence and Engneerng

5 Compacton of the damond T SC graded materal by the hgh speed centrfugal compacton process s the part of the damond partcles settled at the tme t. One can see that ( corresponds to the mass ntensty of the settled damond partcles at the tme t, hence, t equals to the concentraton of the graded compacton produced by centrfugal sedmentaton. An example of W ( for damond-t SC compacts s presented n Fg. 7. a b d [-6 m] d [-6 m] n [%] n [%] Fg. 6. Dstrbuton of the partcles (d, n measured by Shmadzu s apparatus for damond (a and T SC (b W t[s] Fg. 7. Functon W( from ( for damond-t SC compacts damond for the regme of rotaton descrbed n Fg. LetT be the tme when the frst partcles reach the bottom of the vessel. Then the concentraton of settled damond at the tme t s descrbed by the functon l c d ( W ( t T T, ( L wherel denotes the ultmate heght of the graded materal n the vessel.. Analyss of of ynolds and oude and oude numbers The effect of fnte sze of the vessel mples that the lqud s gong to the opposte sde to the fallng partcles wth the average velocty U l ( U (, where U ( s the velocty of partcles, s the concentraton of the partcles n flud. Locally the velocty of the flud can exceed U l (, for nstance for close partcles generatng a cloud []. Polydspersty also can ncrease the local velocty []. Consder the Naver-Stokes equatons whch descrbe quas statonary flows n vscous flud. up( uu g, u, ( where u denote the local velocty, the cnematc vscosty. Dmensonless form of equatons ( can be wrtten as follows [4] u' p' ( u' u' e,, u ', (4 U where l s the number of oude, the unt vector e s gl drected to bottom of the vessel. equently, the gravtatonal forces n equatons (4 are omtted, snce the gravtatonal acceleraton g s small. Ths mples that the oude number s large and the last term n the frst equaton of (4 s less than others. However, for hgh-speed centrfugal rotatons the centrfugal acceleraton s large. Then the oude number becomes small and even for moderate the thrd term n the frst equaton of (4 can exceed other terms. Ths yelds another type of nstablty not related to the nertal terms (ynolds number. Usually the gravtatonal forces serve as a factor ncreasng stablty. In the lnear case of (4 when can be taken as zero, a soluton of the partal dfferental equatons s proportonal to the rato. For the regme presented n Fg. s of order. and s of order 7. Therefore, the vscous flow s lamnar, but the large value of yelds hgh devaton of the local velocty u whch can occur despte of the theoretcal stablty of equatons. However, due to the hgh-speed centrfugal rotaton whch yelds the stratfcaton of the lqud layers by the depth, we have stable stuaton for the Naver-Stokes equatons (4. 4. Expermental nvestgaton of of the graded materals Method of powders consoldaton has a sgnfcant nfluence to mcrostructure and materals propertes. The best compacton s acheved for centrfugal compacton of suspensons whch consst of fne powders (Hgh Speed Centrfugal Compacton Process, HCP. We used the centrfuge UP 65M wth rotatonal speed from Volume 8 Issue November 7 68

6 5 to 5 rpm, for dfferent total tmes of the sedmentaton process. Macrostructure of the FGM damond T SC composte s presented n the Fg. 8. It was obtaned durng 5 s wth constant angular velocty rpm. Acknowledgements Ths work were supported by the Mnstry of Scence and Hgher Educaton of Poland under the project No. N58 /9. ferences Fg. 8. Macrostructure of the FGM damond-t SC composte The measured hardness gradent s presented n Fg. 9 from the top surface to the bottom surface of the sample. hardness HV dstance [mm] Fg. 9. The hardness dstrbuton n the damond composte wth graded mcrostructure 5. Conclusons In ths paper, we construct a mathematcal model of FGM basng on the modfcatons of the Stokes formula. We proposed an algorthm to descrbe sedmentaton of the group of sphercal partcles of dfferent szes and dfferent materals. The results are appled to damond-t SC suspensons to produce graded compacts. The results presented n Fg. 5 demonstrate that for the consdered damond-t SC suspensons the obtaned compact has the structure of the lamnate,.e., the layer of pure damond transforms nto the layer of pure T SC wth the small layer contanng 5% of the both materals. Smulatons show that t s possble to obtan contnuous concentratons of the both materals wth approprate ntal suspensons. Thus, ths method allows to obtan graded materals wth slowly changng of the local propertes. [] R. Borchert, N. Wllert-Porada, An oxdaton resstant metal - ceramc functonally graded materal, Proceedngs of the Conference 9th Cmtec-World Ceramcs Congress, Gettng nto the s Part C, TECHNA, 99, -. [] L. Jaworska, M. Rozmus, B. Królcka, A. Twardowska, Functonally graded cermets, Journal of Achevement n Materals and Manufacturng Engneerng 7 ( [] G. Matula, L.A. Dobrzask, Structure and propertes of FGM manufactured on the bass of HS6-5-, Journal of Achevements n Materals and Manufacturng Engneerng 7 (6-4. [4] M. Mrzababaee, M. Tahan, S.M. Zebarjad, A new approach for the analyss of functonally graded beams, Journal of Achevement n Materals and Manufacturng Engneerng 7 ( [5] L.A. Dobrzask, A. Kloc, G. Matula, J. Domagaa, J.M. Torralba, Effect of carbon concentraton on structure and propertes of the gradent tool materals, Journal of Achevement n Materals and Manufacturng Engneerng 7 ( [6] J. Bandrowsk, H. Merta, J. Zoo, Sedmentaton of suspenton, Slesan Unversty of Technology Press, Glwce,. [7] R.B. Jones, R. Kutteh, Sedmentaton of collodal partcles near a wall, Stokesan dynamc smulatons, Journal of Physcal Chemstry ( [8] M.L. Ekel-Jeewska, B. Metzger, E. Guazzell, Sphercal cloud of pont partcles fallng n a vscous flud, Physcal Fluds 8 (6 -. [9] T. Bosse, L. Kleser, E. Meburg, Small partcles n homogeneous turbulence: Settlng velocty enhancement by two-way couplng, Physcal Fluds 8 (6-8. [] M.P. Brenner, Screenng Mechansms n Sedmentaton, Phys. Fluds ( [] A.J.C. Ladd, Effects of contaner walls on the velocty fluctuatons of sedmentng spheres, Physcal vew Letters 88 ( -4. [] H. Ncola, Y. Peysson, É. Guazzell: Velocty fluctuatons of a heavy sphere fallng through a sedmentng suspenson, Physcal Fluds 8 ( [] E.Guazzell, Evoluton of partcle-velocty correlaton n sedmentaton, Physcal Fluds ( [4] P.J. Mucha, S.Y. Tee, D.A. Wetz, B.I. Shraman, M.P. Brenner, A model for velocty fluctuatons n sedmentaton, Journal of Flud Mechanc 5 ( [5] N.Q. Nguyen, A.J.C. Ladd, Sedmentaton of hard-sphere suspensons at low ynolds number, Journal of Flud Mechanc 55 ( [6] M.C. Bustos, R. Burger, F. Concha, E.M. Tory, Sedmentaton and Thckenng, Kluwer Academc Publ. Dordrecht, 999. [7] S. Ramaswamy, Issues n the statstcal mechancs of steady sedmentaton, Advances n Physcs 5 ( READING DIRECT: