Modeling Crystallization in CMSMPR: Results of Validation Study on Population Balance Modeling in FLUENT

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1 Mdeling Crystallizatin in CMSMPR: Results f Validatin Study n Ppulatin Balance Mdeling in FLUENT Bin Wan and Terry A. Ring Dept. Cheical Engineering Uniersity f Utah Salt Lake City, UT 84 and Kuar M. Dhanasekharan and Jayanta Sanyal Fluent Inc. Caendish Curt Lebann, NH 3766 Abstract Fluent s cputatinal fluid dynaics enirnent has been enhanced with a ppulatin balance capability that perates in cnjunctin with its ultiphase calculatins t predict the particle size distributin within the flw field. The ppulatin balance is sled by ne f the fllwing ethds; discrete ethd, standard ethd f ents, quadrature ethd f ents (QMOM). Fluent s predictin capabilities are tested by using a -diensinal analgy f a cnstant stirred tank reactr with a fluid flw cpartent that ixes the fluid quickly and efficiently using wall eents and has a feed strea and a prduct strea. The results f these Fluent siulatins using QMOM ppulatin balance sler are cpared t steady state analytical slutins fr the ppulatin balance in a stirred tank where ) nucleatin, ) grwth, 3) aggregatin, 4) breakage, take place separately and 5) cbined nucleatin and grwth and cbined nucleatin, grwth and aggregatin takes place. The results f these cparisns shw that the ents f the ppulatin balance are accurately predicted fr nucleatin, grwth, aggregatin and breakage when the flw field is turbulent. With lainar flw the ixing is nt ideal and as a result the steady state well ixed slutins are nt accurately siulated.. Intrductin The ppulatin balance equatin (PBE) is a stateent f cntinuity fr particulate systes. Cases in which a ppulatin balance culd apply include crystallizatin, precipitatin, bubble cluns, gas sparging, sprays, fluidized bed plyerizatin, granulatin, liquid-liquid dispersins, air classifiers, hydrcyclnes, particle classifiers, and aersl flws. In the case f a cntinuus ixed-suspensin, ixed-prduct real (CMSMPR) crystallizer in which aggregatin, breakage and grwth are ccurring the PBE is gien by Randlph and Larsn ( ) as n ( ) n ( ) d( G( ) )) in + b( ) d( ) τ d []

2 with the bundary cnditin, )n. In the abe equatin ) is the nuberbased ppulatin f particles in the tank which is a functin f the particle lue,. The subscript in refers t the inlet ppulatin. G() is the lue dependent grwth rate and b() is the lue dependent birth rate and d() is the lue dependent death rate. In the case f aggregatin the birth and death rate ters are gien by Hulburt and Katz ( ) as / b ( ) d ( ) β ( w, dw ) β (, dw [] a a where the aggregatin rate cnstant, β(,, is a easure f the frequency f cllisin f particles f size with thse f size w. In the case f breakage, the birth and death rate ters are gien by Prasher ( 3 ) as b ( ) d ( ) S( ρ (, dw S( ) ) [3] b b where S() is the breakage rate cnstant that is a functin f particle size,, ρ(, is the daughter distributin functin defined as the prbability that a fragent f a particle f size w will appear at size. It is ften useful t knw the ents f ) because f physical significance. The k th lue ent is defined by k k w dw [4] and represent the ttal nuber and ttal lue f particles in the syste. The PBE can be transfred int a series f ent equatins by ultiplying equatin by k and integrating with respect t fr zer t infinity. These ent equatins are used in place f the PBE t apprxiate the particle size distributin, see Randlph and Larsn ( ). QMOM was first prpsed by McGraw ( 4 ) and further deelped by Marchisi, et. al. ( 5 ). With the QMOM PBE sler in Fluent, a sall nuber f ents, N, (typically 6) are used. Ments are apprxiated by a quadrature apprxiatin that uses N/ weights, W i, and N/ sizes, L i, as fllws: L k N i W L i k i [5] Upn substitutin f these weights and sizes int the N ent equatins, we hae a series f equatins that just equals the nuber f unknwns, N, allwing fr the slutin f the syste f equatins that cnstitutes an apprxiatin f the PBE. Fr the ents the particle size distributin can be recnstituted using a ent transfratin (). Fr re details f this nuerical ethd see the Fluent User s Guide - Crystallizatin Saple Case and Data Files.

3 The ents used in QMOM are length based and are different fr thse described by equatin 4 which are lue based. There is a crrespndence between length based ents and lue based ents. This crrespndence is gien in Table I, where Ka eans surface area cefficient, and K eans lue cefficient. Nting this crrespndence any length based ent calculated by Fluent can be cpared with the lue based ent predicted fr equatin 4 and an analytical slutin t the ppulatin balance. Table Crrespndence between length based and lue based ents Prperty Vlue Based Ment Length Based Ment Nuber f Particles L Surface Area f Particles /3 Ka* L Vlue f Particles K* L 3 In this paper a -D analgue is used t siulate a stirred tank (CMSMPR) within Fluent. These siulatins are cpared t steady-state analytical slutins t the PBE fr ) nucleatin, ) grwth, 3) aggregatin, 4) breakage, take place separately and 5) cbined nucleatin, grwth and aggregatin takes place. The analytical slutins fr ) are cnerted t the length based ents t 5 and cpared directly t the ents predicted by Fluent.. Setup f -D Stirred Tank in Fluent T apprxiate a well ixed stirred tank with a siplified cputatinal geetry we hae used a -D apprxiatin f a stirred tank. This allws testing f the PBE within Fluent siply withut sling a cplicated 3-D flw prble with rtating grids typical f a stirred tank. The -D grid deelped is gien in Figure. It is a square bx. n edge with 4 eleents with an inlet and an utlet bth with a. pening. Fr a cnstant stirred tank, the inlet flw rate is equal t the utlet allwing the ean residence tie t be calculated fr the inlet flw rate (elcity ties inlet area) and the lue (bx area ties unit depth) f the bx. T siulate the agitatin in the tank the tp and btt walls are assued t hae a y-elcity f + and + /s bth in the directin f the utlet fr the turbulent case and -8 and +8 /s in the lainar case. The elcity ectr field fr the lainar case is shwn in Figure and that fr the turbulent case is shwn in Figure 3. The cnectie flux f the tracer at utlet is cllected fr this siulatin and pltted against tie ( 6 ), then cnerted t residence tie distributin using: E () t dc( t) [6] dt The RTD deterined in is way is nralized since the feed tracer cncentratin was..

4 Figure Grid fr -D siulatin f well ixed stirred tank. Figure Velcity distributin fr lainar flw Fluent siulatin. Figure 3 Velcity distributin fr turbulent flw Fluent siulatin. T test the accuracy f the well ixed assuptin, the residence tie distributin was predicted using a unit tracer cncentratin, a secnd phase with the prperties f water, in the tank that is allwed t displace a first water phase as tie prgresses. The utlet cncentratin predicted by the siulatin is shwn in Figure 4 fr the lainar flw and the turbulent flw siulatins as well as the ideal cure. Here we see that the lainar flw cure has an initial peak abe the ideal cure and a tail that is belw the ideal cure. The turbulent siulatin is nearly identical t the ideal cure.

5 . E(t) n lg axis Tie (s) Figure 4 Cparisn f Residence Tie Distributins fr Lainar (black line) and Turbulent (red dts) flw siulatins with Ideal well ixed tank (green line). The ean and standard deiatin f the arius residence tie distributins were deterined giing the fllwing cparisn. t ean /(V/Q) Errr Errr Std.Deiatin/t ean Turbulence del Lainar del Ideal alues fr bth the ean tie, t ean, diided by the rati f tank lue, V, t luetric flw rate, Q and the standard deiatin diided by the ean tie shuld be.. The lainar flw del is clearly wrse than the turbulent flw del in apprxiating an idealized well ixed tank. 3. Nuerical Case Studies Nuerical cases, discussed belw, hae been deelped t test the PBE capabilities f Fluent. First f all Fluent is used t sle the elcity field t a cnergence f -5 fr either the lainar r turbulent flw. Then the a ultiphase calculatin is initiated with the PBE sled by QMOM using 6 length based ents t 5 (r re precisely 3 lengths and 3 weights) with the elcity field fixed. The cnergence criterin is lwered t -7 (r lwer) fr the ultiphase PBE calculatin with a relaxatin paraeter f.9 except when therwise stated. Case -Grwth: The analytical slutin t the PBE, equatin [], fr grwth alne was btained by setting the grwth rate t a cnstant (G()G ), the

6 aggregatin kernel, β(,, and the specific rate f breakage, S(), t zer. Fr this case the feed particle size distributin is n in N ( ) exp( ) [7] and the bundary cnditin is set t zer. n ( ) [8] Althugh nt physically realistic, a cnstant grwth rate results in a PBE slutin is n ( ) : exp G τ ( ) N + + exp + G τ G τ G τ [9] where τ is the ean residence tie. Siulatins are perfred with N and set equal t unity. Please nte that this slutin is singular when τ G. Analytical expressins ( 7 ) fr the zerth, first and secnd lue based ents are, in, in, in + τ G + τ G [] These and ther ent equatins are used fr cparisn with Fluent siulatins. Because in the analytical slutin, the grwth ter is expressed in ters f the particle lue and in FLUENT it needs t be expressed in ters f particle length, x: d( x) G( ( x)) dt [] 3 ( x) K x [] 3 d( x) d( K x ) dx 3Kx 3Kx GL ( x) [3] dt dt dt Where K eans lue cefficient, fr S, in FLUENT, we use

7 G( ( x)) GL ( x) [4] 3K x Fr the Fluent siulatin the grwth rate f µ/s and the ean residence tie f s was used with the initial particle size distributin in the tank gien by the feed distributin, the siulatin tk 6 iteratins t achiee a cnergence criterin f -9 fr the turbulent flw siulatin with a relaxatin factr f. and 4 iteratins t achiee a cnergence criterin f -9 fr the lainar flw siulatin. The results f these siulatins are gien in Table fr bth the lainar and turbulent flw siulatins with a relaxatin factr f.9. The turbulent flw case will nt cnerge with a relaxatin factr f.. The results f the turbulent flw siulatin are accurate t nly. with this cnergence criterin. The results f the lainar flw siulatin is less accurate a 3.7 errr with the sae cnergence criterin. Because the lainar flw siulatin des nt crrespnd t well-ixed cnditins, it des nt accurately siulate the analytical slutin. Table Ment cparisn f PBE fr Fluent Siulatins with Analytical Slutin fr Grwth Turbulence Lainar Inlet Outlet (Analytical) Outlet (Fluent) Errr Outlet (Fluent) Errr L L L L L e L e Case -Nucleatin and Grwth: The analytical slutin t the PBE, equatin [], fr nucleatin and grwth was btained by setting the grwth rate t a cnstant (G()G ), the aggregatin kernel, β(,, and the specific rate f breakage, S(), t zer. Fr this case the feed particle size distributin is set t zer, n in () and the bundary cnditin is n ( ) n [5] where n is the nuber density f particles with a zer size. The nucleatin rate is gien by the prduct f G and n. The analytical slutin fr this case is gien by ( ) ) n exp( ) [6] Gτ

8 This analytical slutin is cnerted t length based ents fr cparisn with the Fluent siulatin. The Fluent siulatin was run with G L-. /s nting the abe cnersin in equatins t4 and the nucleatin rate is #/ 3 /s, the ean residence tie, τ, f s with n particles in the feed. The results f this cparisn are gien in Table 3. Here we see that the lainar flw siulatin is in errr by as uch as ~5 while the turbulent flw siulatin is accurate t ~.. Table 3 Ment cparisn f PBE fr Fluent Siulatins with Analytical Slutin fr Nucleatin and Grwth. Analytical FLUENT FLUENT Slutin Lainar Errr Turbulence Errr L L L L L L Case 3-Aggregatin: The analytical slutin t the PBE, equatin [], fr aggregatin alne was btained by setting the grwth rate t zer (G()), the aggregatin kernel t a cnstant, β(, β, and the specific rate f breakage, S(), t zer. Fr this case the feed particle size distributin is set t an expnential distributin gien by equatin 7. The analytical slutin fr this case is gien by ( 8 ) n ( ) N I ( β N τ) β N τ ( ) + I + ( β N τ) ( + β + ( β N τ) N τ) exp + ( β N τ) + β N τ ( ) [7] where I (z) and I (z) are dified Bessel Functins f the first kind f zer and first rders. This analytical slutin is cnerted t length based ents fr cparisn with the Fluent siulatin. Analytical expressins ( 9 ) fr the zerth, first and secnd lue based ents are +, in, in + β β τ + τ β, in τ [8]

9 The Fluent siulatin was run fr the cnditins f β, N,, and a ean residence tie f s. The turbulent PBE siulatin ran fr 5 iteratins t get t a residue f -7. The results f this cparisn are gien in Table 4 fr the turbulent flw case nly. Here we see that the turbulent flw siulatin is accurate t ~.4 and crrectly predicts that the 3 rd length based ent is crrectly predicted t nt change during passage thrugh the reactr. Table 4 Ment Cparisn f PBE fr Fluent Turbulent Siulatins with Analytical Slutin fr Aggregatin Alne. Inlet Outlet (Analytical) Outlet (Fluent) Errr L L L L L L Case 4-Breakage: The analytical slutin t the PBE, equatin [], fr breakage alne was btained by setting the grwth rate t zer (G()), the aggregatin kernel t zer, β(,, the specific rate f breakage t S() and the daughter distributin functin is set t ρ(,/w. Fr this case the feed particle size distributin is set t an expnential distributin gien by equatin 7. The analytical slutin fr the case is gien by ( 7 ): N n Breakage (, τ) ( + τ ) + τ + τ ( + ) : ( + τ ) 3 exp [9] This analytical slutin is cnerted t length based ents fr cparisn with the Fluent siulatin. Analytical expressins f the zerth and first lue ents can be deried t gie τ +, in [] which indicate that the lue f particles is cnsered. These ents are cnerted t length based ents fr direct cparisn with the QMOM Fluent siulatin.

10 The Fluent siulatin was run with cnstants N and were set t unity, the ean residence tie, τ, f s. Cparisn the Fluent siulatin t the analytical slutin fr the turbulent flw siulatin is gien in Table 5. Using a cnergence criterin f - required iteratins using a relaxatin factr f.9. The results fr L 3 are accurately predicted indicating that ass is cnsered, and the errr fr ther ents are within 4.9. This indicates that Fluent QMOM des accurately siulate breakage. Table 5 Ments Cparisn f FLUENT Turbulent Siulatin t the Analytical slutin fr Breakage nly Inlet Outlet (Analytical) Outlet (Fluent) Errr L L L L E-5 L L Case 5-Nucleatin, Grwth and Aggregatin Cbined: The analytical slutin t the PBE, equatin [], fr nucleatin, grwth and aggregatin tgether was btained by setting the grwth rate t a cnstant (G()G ), the aggregatin kernel t a cnstant, β(, β, and the specific rate f breakage, S(), t zer. Fr this case the feed particle size distributin is set t zer, n in () and the bundary cnditin is n ( ) n [3] where n is the nuber density f particles with zer size. The nucleatin rate is gien by the prduct f G and n. The analytical slutin fr this case is gien by ( ) exp + β n G β n G τ G n ( ) n I G β n G G β n G [4] where I (z) is the dified Bessel Functin f the first kind f first rder. Analytical expressins f the zerth, first and secnd lue ents can be deried t gie

11 + τ G τg + β n G τ β τ + τ β [5] This analytical slutin and the abe ent equatins are cnerted t length based ents fr cparisn with the Fluent siulatin. The Fluent siulatin was run with G -. 3 /s and the nucleatin rate is #/ 3 /s, the ean residence tie, τ, f s with n particles in the feed. The slutin tk 3 iteratins t reach a cnergence criterin f -7. The results f this cparisn are gien in Table 6. The largest errr is.4 in the length ent, L. Table 6 Ments Cparisn f FLUENT Turbulent Siulatin t the Analytical Slutin t the PBE fr Nucleatin, Grwth and Aggregatin Outlet (Analytical) Outlet (Fluent) Errr L L L L L L Cnclusins The -D del f a well-ixed stirred tank is a siple geetry with a sall nuber f grids can be shwn t be an accurate del if the flw is turbulent. Using this turbulent del f a well-ixed tank, a tw phase del with a PBE fr the secnd, slid phase has been deelped and sled with the QMOM ptin in Fluent. This del has been tested using nuerical cases where grwth, aggregatin, breakage and the cbined cases f nucleatin and grwth and nucleatin grwth and aggregatin. These Fluent siulatins are cpared with analytical slutins t the PBE fr a cnstant well-ixed tank fr these cases. The QMOM ptin in Fluent accurately predicts each f these cases. T btain less than accuracy fr these cases, different cnergence criterin are necessary. Depending upn the case, a cnergence criterin between -7 t -4 is required. References Randlph, A.D. and Larsn,M.A., Thery f Particulate Prcesses, nd ed. 988.

12 Hulburt, H.M. and Katz, S. Se Prbles in Particle Technlgy Statistical Mechanical Frulatin, Che. Eng. Sci. 9,555,(964). 3 Prasher, C.L., Crushing and Grinding Prcess Handbk, Wiley, New Yrk (987). 4 McGraw, R., Descriptin f Aersl Dynaics by Quadrature Methd f Ments, Aersl Science and Technlgy, 7,55-65,(997). 5 Marchisi, D.L., Virgil, R.D. and Fx, R.O., Quadrature Methd f Ments fr Aggregatin-Breakage Prcesses, J. Cllid and Interface Sci. 58,3-334,(3). 6 Byung S. Chi, Bin Wan, Susan Philyaw, Kuar Dhanasekharan, and Terry A. Ring, Residence Tie Distributins in a Stirred Tank: Cparisn f CFD Predictins with Experient, Ind. Eng. Che. Res.; 4; 43() pp Nicanis, M. and Hunslw, M.J., Finite-eleent Methds fr Steady-State Ppulatin Balance Equatins, AIChE J., 44(),58-7,(998). 8 Hunslw, M.J., A Discrete Ppulatin Balance fr Cntinuus Systes at Steady State, AIChE J. 36(), 6 (99). 9 Sit, D. J., M. J. Hunslw, and W. R Patersn, Aggregatin and Gelatin : Analytical Slutins fr CST and Batch Operatin, Che. Eng. Sci, 49(7), 5 (993) Lia, P.F. and Hulburt, H.M., Aggleratin Prcesses in Suspensin Crystallizatin, AICHE Meeting, Chicag (Dec. 976).