Q & Particle-Gas Multiphase Flow. Particle-Gas Interaction. Particle-Particle Interaction. Two-way coupling fluid particle. Mass. Momentum.

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1 Paicle-Gas Muliphase Flow Fluid Mass Momenum Enegy Paicles Q & m& F D Paicle-Gas Ineacion Concenaion highe dilue One-way coupling fluid paicle Two-way coupling fluid paicle Concenaion highe Paicle-Paicle Ineacion collision conac

2 Discee paicle simulaion Paicle Mico Fluid Meso dy dx

3 Discee paicle simulaion () Collision fee flow (dilue phase flow) One-way () Collision dominaed flow (inemediae concenaion) Two-way (3) Conac dominaed flow (dense phase) Fluid : negleced Fluid : aken ino accoun One-way Two-way

4 Paicle-Paicle Ineacion. Collision fee flow Dilue phase flow. Collision dominaed flow Dispesed flow 3. Conac dominaed flow Dense phase flow

5 Collision Dominaed Flow

6 Cluse : Cloud, Heeogeneous concenaion Rise cluse 00[mm] Ciculaing fluidized bed Tubulence of cluse scale Lage influence on anspo phenomena

7 Dispesed flow (collision-dominaed) Paicle velociy v Fluid velociy u lif Angula velociy ω dag v s Tanslaion : v s + & 0 x 0 Δ Fluid foce (dag & lif) & f F x + g m mass gaviy x x + vs Δ Gaviy acceleaion veco Roaion: &ω T I 0 ω ω 0 + ω 0 Δ & Toque Ineial momen ( ) πρ s a

8 Had sphee model Impulsive equaions ( 0) m ( V V ) J ( 0) m ( V V ) J ( 0) I( ω ω ) n J ( 0) I ( ω ω ) n J (0) V, ω (0) (0) V, ω (0) V, ω V, ω n J I Nomal uni veco dieced fom Paicle o Impulsive foce exeed on Paicle Momen of ineia I ( / 5) m

9 G G Relaion beween pe-and pos- collision velociies n (0) (0) G c (0) c n G G (0) f + c (0) f + c e : coefficien of esiuion f : ficion coefficien (0) (0) V V Relaive velociy beween paicle cenes (0) (0) (0) G + Relaive velociy of he conac poin ω n + ω n (0) (0) G Tangenial componen of elaive velociy c ( G n)n (0) Gc : Nomal uni veco dieced fom Paicle o (0) : Tangenial uni veco Gc (0) (0) n G < > 7 ( e) G 7 ( e) ( 0) ( 0) m V V ( n f)( n G )( + e) m + m ( 0) ( 0) m V V + ( n f)( n G )( + e) m + m ω ω 5 m m + m (0) (0) ω ( n G )( n ) f ( + e) 5 m m + m (0) (0) ω ( n G )( n ) f ( + e) ( 0) ( 0) ( 0) m V V ( + e)( n G ) n+ Gc 7 m + m ( 0) ( 0) ( 0) m V V + ( + e)( n G ) n+ Gc 7 m + m ( 0) 5 ( 0 ω ω ) ( ) 7 G n m c m + m ( 0) 5 ( 0 ω ω ) ( ) 7 G n m c m + m

10 Collision-dominaed Flow Equaions of moion Relaionship beween pe- and pos collision velociies Finding collision panes How? Deeminisic Taecoies of all paicles Pinciple of sepaaion Sochasic (DSMC) Diec Simulaion Mone Calo Pobabiliy Taecoies of sample paicles

11 Deeminisic mehod based on aecoies No collide k ( ) +Δ a ( ) + k + Δ +Δ + Δ a + Δ Collide k ( ) +Δ ( ) + k + Δ a + Δ a + Δ ( ) ( ) + k a + a + d a if k has a soluion 0<k<, paicles collide.

12 Deeminisic vs DSMC Deeminisic Taecoies of all paicles Sochasic (DSMC) Taecoies of sample paicles Diec Simulaion Mone Calo Pobabiliy

13 Sochasic (DSMC) mehod Field of physical paicles Sample paicles Field eplaced wih sample paicles

14 Collision pobabiliy Collision pobabiliy of Paicle i colliding Paicle Numbe of sample paicle : N Numbe densiy : Relaive velociy : n G i v i i v Numbe densiy of sample paicle G i Δ D p n s n N The numbe of paicles in he ube : P i n π D s p G δ i

15 DSMC Decision No collision Find collision pane Decision Collision Find collision pane Decision No collision Find collision pane

16 In DSMC mehod, calculaion of aecoies ae made only fo sample paicles. When calculaing aecoies of each sample paicle in ime sep Δ, you have o conside whehe a sample paicle collides ohe paicles which ae epesened by anohe sample paicle. Numbe of sample paicle N Numbe of ue paicles n Numbe of sample paicle n s n/n If he sample paicle i collides necessaily wih any one of paicles in he field, he collision pobabiliy ha he sample paicle i collides wih any one of paicles in he field is. In his case, he collision pobabiliy ha he sample paicle i collides wih one of paicles epesened by sample paicle is /N, because each sample paicle has he equal possibiliy o collide wih he sample paicle i. N + N + + N + + N

17 Collision pobabiliy ha he sample paicle i collides wih one of paicles epesened by sample paicle unde he condiion ha he sample paicle i collides necessaily wih any one of paicles in he field N + N + + N + + N sample paicle i vs. sample paicle. sample paicle i vs. sample paicle. sample paicle i vs. sample paicle N. N N P i N N The condiion ha he sample paicle i collides necessaily wih any one of paicles in he field Acual collision pobabiliy P i ha he sample paicle i collides wih one of paicles epesened by sample paicle No saisfied P i < N

18 Collision pobabiliy Paicle i o Paicle P i n s π D p G i Δ 0 < P i < /N Paicle i o Paicle P i n s π D p G i Δ 0 < P i </N Paicle i o Paicle P i n s π D p G i Δ 0 < P i < /N Paicle i o Paicle N P in n s π D p G in Δ 0< P in </N R R 0 P i P i P i /N /N (-)/N /N (N-)/N P in Selec a candidae of collision pane Jin[R N]+, R :andom numbe if R > (/N) - P i, Paicle i collides Paicle. if R < (/N) - P i, Paicle i does no collide Paicle.

19 Example calculaion Le us conside i, i.e., conside Paicle Le he numbe of sample paicles be N000 3 Le a andom numbe fom a geneao R is calculaed, in[r N]+in[ ]+35 Paicle 35 is he candidae of he collision pane of Paicle 5 Collision pobabiliy P 35 is calculaed fom P i n s π D p G i Δ Fo example, le P 35 be P /N - P i 35/ Compaing he andom numbe R wih he above value, 8 R < (/N) - P i Theefoe, Paicle does no collide wih Paicle 35. R Pi (-)/N /N

20 Teamen of of fluid moion Navie-Sokes o Eule equaion Meso-Scale dy Local aveaged dx

21 Equaions of fluid moion Equaion of coninuiy ε ( ε u + x Equaion of moion (inviscid) ) 0 u ε p ρ f pi : velociy : void facion : pessue : fluid densiy : paicle-fluid ineacion Dag f β v u ) pi ( i i ( ε u u i ) ( ε iu ) + x ε p ρ x i + f si

22 Collision Dominaed Flow Cluse Fomaion D 3D

23 Tanaka e al.(993) In-elasic collision Cluse fomaion

24 3D 3D simulaion Tsui s laboaoy (998)

25 Mixe Mixe Avalanche Hou glass

26 Fluidized bed Pneumaic conveying

27 Conac-dominaed flow Foces acing on paicles? Paicle velociy v Roaion ω dag Fluid velociy u Conac foces (nomal and angenial) gaviy

28 Sof sphee model Defomaion Ovelap

29 Compession peiod () Recovey peiod ()

30 Damping vibaion k : siffness m : mass η : damping coefficien x 0 + Equaion of moion : mass acceleaion foce m & x η x& k x Viscous Damping Resoing foce due o linea sping

31 x k x x m & & & η x e x λ Subsiuing o he above equaion k m ηλ λ > 4km η e C C e x λ λ + m k m m + η η λ m k m m η η λ 4km η ( ) m e C C x η + η < 4km + m m k C m m k C e x m 4 sin 4 cos η η η + ε η η m m k Ce x m 4 cos

32 m & x + x& Iniial condiion: η + kx 0 0 : x x x & v 0, 0 Sol. x Ce η m cos k m η 4m + ε x T ηm Ce x 0

33 m & x + η x& + kx 0 Iniial condiion: 0 : x 0, x& v 0 v0 x exp γ ω sin q + x q v x& exp γ ω q cos q γ q Sol. ( ) ( ) 0 ( ){ ( ) sin( q ) } 0 ω x x 0 whee T kn ηn ω, γ, q ω γ m mk 0 Vibaion peiod T π q

34 Sof sphee model Paicle conac foce: f f + f C Cn C nomal foce angenial foce δ C slide δ n C Paicle i Sping : Siffness Young s modulus, Poisson aio Paicle i Dash-po : Damping coefficien Coefficien of esiuion

35 Sof sphee model sping slide Dash-po coupling (a) nomal foce conac foce : fc fcn + fc (b) angenial foce nomal foce Hez angenial foce Mindlin Sping : Siffness Young s modulus, Poisson aio Dash-po : Damping coefficien Coefficien of esiuion

36 Nomal Foce Paicle adius i Paicle i δ n f Cni Paicle adius η ( ) 3/ k n δ ni n v i n i n i Paicle Siffness Ovelap Damping coefficien Velociy veco of paicle I elaive o paicle Uni veco dawn fom paicle i o paicle Poisson aio k n 4 3 ν Ei i ν + E i + i Young s modulus

37 ( ) i i i n ni n Cni n n v k f η δ 3/ i i i i n E E k ν ν Nomal Foce Nomal Foce ( ) 3 + i i p p n E k ν Conac beween sphees wih he same physical popeies and diffeen adiuses p i p i E E E, ν ν ν Conac beween sphee and wall p w w p p n E E k ν ν Conac beween sphees wih he same physical popeies and he same adiuses p i ( ) 3 p p p n E k ν

38 Tangenial Foce Paicle adius i δ Tangenial defomaion Paicle i δ n f Ci Paicle adius k δ η v si Paicle Siffness Tangenial defomaion Damping coefficien Tangenial componen of elaive velociy a conac poin Poisson aio k ν i δ n G i ν i Gi Laeal Young s modulus

39 si Ci v k f δ η Tangenial Foce Tangenial Foce 8 n i i i i G G k δ ν ν + + Conac beween sphees wih he same physical popeies and diffeen adiuses p i p i G G G, ν ν ν 4 n i i p p G k δ ν + Conac beween sphee and wall 8 n w w p p p n G G k δ ν ν + Conac beween sphees wih he same physical popeies and he same adiuses p i n p p p G k δ ν ( ) +ν E G

40 Equaions of of paicle moion f F fc Tanslaion : & x + m + g Roaion: &ω T I x m g T : Posiion veco :Angula velociy : Mass f F : Fluid foce (dag & lif) : Gaviy acceleaion veco f C : Conac foce :Toque v ω I : Ineial momen ( ) 8 5 πρ s a 5 v s v s + & 0 x 0 Δ x x + vs Δ 0 ω ω 0 + ω 0 Δ &

41 (x, y ) (x, y ) (x, y ) (xi, yi) (xi, yi) δ n (xi, yi) δ n (x, y ) (x, y ) (x, y ) (xi, yi) (xi, yi) δ n δ n (xi, yi)

42 Teamen of of fluid moion Navie-Sokes o Eule equaion Local aveaged Finie diffeence mehod dy dx

43 Equaions of fluid moion Equaion of coninuiy ε ( ε u + x Equaion of moion (inviscid) ) 0 u ε p ρ f pi : velociy : void facion : pessue : fluid densiy : paicle-fluid ineacion Dag f β v u ) pi ( i i ( ε u u i ) ( ε iu ) + x ε p ρ x i + f si

44 Paicle-fluid ineacion em Dag f β v u ) pi ( i i ε < 0.8 μ β D ε ( ε ) [ 50( ε ).75 Re ] + D Re D ρ α u v D c μ ε > 08. β 3 4 ρ( ε) ε. C v u D Dp 7 C D ( Re ) Re LL(Re 000) 0. 43LLLLLLLLL(Re > 000) Re D u / ν p

45 Calculaion of of fluid moion (SIMPLE mehod) (Semi-Implici Mehod fo Pessue-Linked Equaion) Saggaed gid : Gid fo scala (pessue, void facion) Gid fo veco (velociy) Sa Disceizaion : finie diffeence Assume pessue Soluion algoihm : Modify pessue + Δ no Calculae velociy fom momenum equaion Check whehe he equaion of coninuiy is saisfied yes Daa save End

46 Main flow Sa iniialize all daa +Δ calculae void facion calculae fluid moion calculae paicle moion save daa end

47 Paicle moion Calculae fluid moion Does he paicle conac ohe paicle (wall)? Yes No Repea N imes N:paicle numbe Calculae conac foce acing on he paicle Calculae Paicle Moion Calculae fluid foce acing on he paicle Calculae paicle acceleaion and velociy Calculae paicle posiion Save daa

48 Examples of of DPS () Collision fee flow (dilue phase flow) () Collision dominaed flow (inemediae concenaion) Tubulen diffusion Dus collecion Dilue phase paicle anspo ( pneumaic o hydaulic) Ciculaing fluidized bed Paicle anspo ( pneumaic o hydaulic) (3) Conac dominaed flow (dense phase) Fluid : negleced Fluid : aken ino accoun Soil mechanics Avalanche Simple shea flow Vibaing bed Hoppe & chue flow Scew feede Ball mill Roaing dum Mixe Tumbling Ganulao Copy machine Compacion Cush Dampe Sieve Fluidized bed Dense phase anspo Colloid

49 Tanaka, Ishida and Tsui, (99) Flow ime Dense phase pneumaic conveying

50 Roaing dum Compaison wih measuemens Non-spheical paicle Segegaion

51 Cenifugal umbling ganulao Compaison wih measuemens Cohesion foce: Effec of liquid bidge

52 D fluidized Bed

53 3D Fluidized bed

54 Paicle numbe : Paicle size Similaiy paamee: k M. Kazai, K. Roko, T. Kawaguchi, T. Tanaka and Y. Tsui: A Sudy on Condiions fo Similaiy of Paicle Moion in Numeical Simulaion of Dense Gas-Solid Two Phase Flow, ICMF 95 Kyoo (995) D D p p0 : Similaiy paamee Paicle densiy Fluid viscosiy Fluid dag ρ0 ρ μ kμ 3 0 k β β0 3 k Coecion of void facion : Face Ceneed Cubic sucue: M. Sakano, T. Minoua and H. Nakanishi : Numeical Simulaion of Twodimensional Fluidized Bed Using Discee Elemen Mehod, 7h Symposium on Muliphase Flows, Tokyo (995)

Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations

Today - Lecture 13. Today s lecture continue with rotations, torque, Note that chapters 11, 12, 13 all involve rotations Today - Lecue 13 Today s lecue coninue wih oaions, oque, Noe ha chapes 11, 1, 13 all inole oaions slide 1 eiew Roaions Chapes 11 & 1 Viewed fom aboe (+z) Roaional, o angula elociy, gies angenial elociy