Lecture 17 - Eulerian-Granular Model. Applied Computational Fluid Dynamics

Transcription

1 Lectue 7 - Euleian-Ganula Model Applied Computational Fluid Dynamic Intucto: Andé Bakke Andé Bakke (00-006) Fluent Inc. (00)

2 Content Oveview. Deciption of ganula flow. Momentum equation and contitutive law. Intephae exchange model. Ganula tempeatue equation. Solution algoithm fo multiphae flow. Example.

3 Oveview The fluid phae mut be aigned a the pimay phae. Multiple olid phae can be ued to epeent ize ditibution. Can calculate ganula tempeatue (olid fluctuating enegy) fo each olid phae. Calculate a olid peue field fo each olid phae. All phae hae fluid peue field. Solid peue contol the olid packing limit.

4 Ganula flow egime Elatic Regime Platic Regime Vicou Regime. Stagnant Slow flow Rapid flow Ste i tain Stain ate Stain ate dependent independent dependent Elaticity Soil mechanic Kinetic theoy

5 Kinetic theoy of ganula flow Kinetic Kinetic Tanpot Tanpot Colliional Colliional Tanpot Tanpot

6 Ganula multiphae model: deciption Application of the kinetic theoy of ganula flow Jenkin and Savage (983), Lun et al. (984), Ding and Gidapow (990). Colliional paticle inteaction follow Chapman-Enkog appoach fo dene gae (Chapman and Cowling, 970). Velocity fluctuation of olid i much malle than thei mean velocity. Diipation of fluctuating enegy due to inelatic defomation. Diipation alo due to fiction of paticle with the fluid.

7 Ganula multiphae model: deciption () Paticle velocity i decompoed into a mean local velocity and a upeimpoed fluctuating andom velocity. A ganula tempeatue i aociated with the andom fluctuation velocity: u C 3 θ v C C

8 Ga molecule and paticle diffeence Solid paticle ae a few ode of magnitude lage. Velocity fluctuation of olid ae much malle than thei mean velocity. The kinetic pat of olid fluctuation i aniotopic. Velocity fluctuation of olid diipate into heat athe fat a a eult of inte paticle colliion. Ganula tempeatue i a bypoduct of flow.

9 Analogy to kinetic theoy of gae Velocity ditibution function Pai ditibution function Colliion ae bief and momentaily. No intetitial fluid effect. Fee teaming Colliion

10 Ganula multiphae model: deciption Seveal tanpot mechanim fo a quantityψ within the paticle phae: Kinetic tanpot duing fee flight between colliion Requie velocity ditibution function f. Colliional tanpot duing colliion Requie pai ditibution function f. Pai ditibution function i appoximated by taking into account the adial ditibution function g o (, σ ) into the elation between and f and f.

11 Applying Enkog kinetic theoy fo dene gae give fo: Continuity equation fo the ganula phae. Ganula phae momentum equation. f m u t & + ) ( ) ( ρ ρ f n f f f F u m R p u u u t & ) ( ) ( ) ( τ ρ ρ Fluid peue Solid te teno Phae inteaction tem Ma tanfe Continuity and momentum equation

12 Contitutive equation Contitutive equation needed to account fo intephae and intaphae inteaction: Solid te Account fo inteaction within τ olid phae. Deived fom ganula kinetic theoy τ P I whee, S P g o λ, µ T ( u + ( u ) ) + µ S + ( λ 3 Stain µ ) u Solid Radial Solid ate Peue ditibution bulk I and function hea vicoity

13 Contitutive equation: olid peue Peue exeted on the containing wall due to the peence of paticle. Meaue of the momentum tanfe due to teaming motion of the paticle: P ρθ ( ω + ( + e ) go ) Gidapow and Syamlal model: ω Sinclai model: ω ( + d 6 D )

14 The adial ditibution function g 0 ( ) i a coection facto that modifie the pobability of colliion cloe to packing. Expeion fo g 0 ( ): Ding and Gidapow, Syamlal et al. Sinclai. 0.65, ) ( max, 3,max o g ) ( 3 ) ( g o + Contitutive equation: adial function

15 Contitutive equation: olid vicoity The olid vicoity: Shea vicoity aie due tanlational (kinetic) motion and colliional inteaction of paticle: η + ) / Colliional pat: Gidapow and Syamlal model: Sinclai model: µ + µ µ, coll, kin 8 5 θ π µ, coll ρdgoη ( µ, col 5dρ ( θ π ) o ( ) η(3η ) go η g η 5 5π

16 Contitutive equation: olid vicoity Kinetic pat: Syamlal model: dρ ( θ π ) 8 µ, kin + η(3η ) g ( η) 5 o Gidapow model: 5d ρ ( θ π ) 8 + go η 96ηg 5 µ, kin o Sinclai model: 5d ρ ( θ π ) 8 µ ω + η(3η ) g, kin 96 η( η) g 5 o o

17 Contitutive equation: bulk vicoity Bulk vicoity account fo eitance of olid body to dilatation: λ 4 ρ d 3 g o ( + e θ ) π e d volume faction of olid. coefficient of etitution. paticle diamete.

18 Platic egime: fictional vicoity In the limit of maximum packing the ganula flow egime become incompeible. The olid peue decouple fom the volume faction. In fictional flow, the paticle ae in enduing contact and momentum tanfe i though fiction. The tee ae detemined fom oil mechanic (Schaeffe, 987). The fictional vicoity i: P inϕ µ, fict I The effective vicoity in the ganula phae i detemined fom the maximum of the fictional and hea vicoitie: [ µ µ ] max, coll, kin, µ, fict µ +

19 Momentum equation: intephae foce Inteaction between phae. n ( R ) 0 Fomulation i baed on foce on a ingle paticle coected fo effect uch a concentation, cluteing paticle hape and ma tanfe effect. The um of all foce vanihe. Dag: caued by elative motion between phae; K f i the dag between fluid and olid; K l i the dag between paticle n ( K l l ( u l f u + m& f u )) + K f f ( u f u ) 0 Geneal fom fo the dag tem: K f ρ f τ dag f With paticle elaxation time: τ f ρd 8µ f

20 Momentum: intephae exchange model Fluid-olid momentum inteaction, expeion fo f dag. Aatopou et al (990). Di Felice (994). Syamlal and O Bien (989). Wen and Yu (966). Dag baed on Richadon and Zaki (954) and/o Egun (95). ue the one that coectly pedict the teminal velocity in dilute flow. in bubbling bed enue that the minimum fluidized velocity i coect. It depend tongly on the paticle diamete: coect diamete fo non-pheical paticle and/o to include cluteing effect.

21 Compaion of dag law A compaion of the fluid-olid momentum inteaction, f dag, fo: Relative Reynold numbe of and 000. Paticle diamete 0.00 mm. f Syamlal-O'Bien Schuh W en-yu Di Felice Aatopoo f Syamlal-O'Bien Schuh W en-yu Di Felice Aatopoo Ganula volume faction Ganula Volume Faction

22 ) ( ) ( ) 8 )( 3( 3 3 m l m m l l olm m l m m l l lm lm lm u u d d g d d C e K ρ ρ π ρ ρ π π + + M k k k m l f l m f olm d d d d d g ) ( 3 Paticle-paticle dag law Solid-olid momentum inteaction. Dag function deived fom kinetic theoy (Syamlal et al, 993).

23 Momentum: intephae exchange model Vitual ma effect: caued by elative acceleation between phae Dew and Lahey (990). u f u K C ρ ( + u u ) ( + u u ) vm, f vm f f f t t Lift foce: caued by the heaing effect of the fluid onto the paticle Dew and Lahey (990). K C ρ ( u u ) ( u k, f L f f f Othe intephae foce ae: Baet Foce, Magnu Foce, Themophoetic Foce, Denity Gadient Foce. )

24 Ganula multiphae model: ma tanfe Unidiectional ma tanfe: m& f Define poitive ma flow i pecified contant ate of ate pe unit volume fom phae f to phae, m& popotional to: & ρ f paticle hinking o welling. e.g., ate of buning of paticle. Heat tanfe modeling can be included via UDS. f

25 Ganula tempeatue equation Ganula tempeatue. 3 θ C C 3 { ( ρ θ ) + ( ρ u θ )} τ : u t Diffuion tem +... Poduction tem + ( κ θ θ ) γ + φ + φ lm f Diipation tem due to inelatic colliion Exchange tem

26 Contitutive equation: ganula tempeatue Ganula tempeatue fo the olid phae i popotional to the kinetic enegy of the andom motion of the paticle. epeent the geneation of τ : u enegy by the olid te teno. κ θ θ ) epeent the diffuion of enegy. ( κ θ Ganula tempeatue conductivity.

27 Contitutive equation: ganula tempeatue Ganula tempeatue conductivity. Syamlal: 5 ρd θ π κ [ + η θ g o (4η 3) 4(4 33η ) 5 Gidapow: κ θ 75ρd θ π [ 384ηg + o g 5 o η] 6 + (4 33η ) η g o ] 5π + ρ d ( + e ) g o θ π Sinclai: κ θ ( κ ) θ yamlal + 5ρ d 6ηg θ π + η (4η 3) g ω η o o

28 Contitutive equation: ganula tempeatue γ θ epeent the diipation of enegy due to inelatic colliion. 4 θ Gidapow: γ θ 3( e ) ρgoθ u π π Syamlal and Sinclai: γ θ ( e ) g d π o ρ θ 3 Lun et al (984) Hee φ lm epeent the enegy exchange among olid phae (UDS).

29 Contitutive equation: ganula tempeatue φ f epeent the enegy exchange between the fluid and the olid phae. Lamina flow: φ 3K θ f f Dipeed tubulent flow: Sinclai: φ f K f ( k f 3θ k f ) Othe model: φ f ' ' K f (k f < u pi, u fi > )

30 Tet cae fo Euleian ganula model Contou of olid team function and olid volume faction when olving with Euleian-Euleian model. U7 m/ Solid% Contou of olid team function and olid volume faction when olving with Euleian-Ganula model.

31 Solution guideline All multiphae calculation: Stat with a ingle-phae calculation to etablih boad flow patten. Euleian multiphae calculation: Copy pimay phae velocitie to econday phae. Patch econday volume faction() a an initial condition. Fo a ingle outflow, ue OUTLET athe than PRESSURE-INLET; fo multiple outflow boundaie, mut ue PRESSURE-INLET. Fo ciculating fluidized bed, avoid ymmety plane (they pomote unphyical clute fomation). Set the fale time tep fo undeelaxation to Set nomalizing denity equal to phyical denity. Compute a tanient olution.

32 Summay The Euleian-ganula multiphae model ha been decibed in the ection. Sepaate flow field fo each phae ae olved and the inteaction between the phae modeled though dag and othe tem. The Euleian-ganula multiphae model i applicable to all paticle elaxation time cale and Include heat and ma exchange between phae. Seveal kinetic theoy fomulation available: Gidapow: good fo dene fluidized bed application. Syamlal: good fo a wide ange of application. Sinclai: good fo dilute and dene pneumatic tanpot line and ie.