Modeling of Breakage Equations Using PBE Software

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1 Modelng of Breakage Equatons Usng PBE Software Prsclla Hll 1 Abstract Partcle breakage occurs n a varety of processng unts n chemcal processng rangng from crushers and grnders to breakage of partcles n strred vessels. Changes n the partcle sze dstrbuton due to breakage are most often accounted for usng populaton balance equatons (PBEs). These PBEs are partal ntegral dfferental equatons that are dffcult to solve and for whch there are few analytcal solutons. Ths often makes solvng these equatons ntractable for undergraduate students. The goal s to remove ths barrer so that students can focus on the effects of breakage model parameters on the resultng partcle sze dstrbutons. An ntal verson of a computer program has been developed for use n a partcle and crystallzaton technology electve course n chemcal engneerng. The program uses MS Excel as an nterface and allows the user to set the ntal partcle sze dstrbuton and the model parameters. Keywords: Populaton Balance Equatons, Software. INTRODUCTION In addton to standard lectures, the course wll encourage actve learnng. Research shows [1-4] that students learn more when they are actvely nvolved n the learnng process. Wankat [4] states that teachng technques such as computer smulatons ncrease student learnng because the students are actvely nvolved and tend to spend more tme on learnng. To facltate the comprehenson of PBE concepts, educatonal software wll be developed for use n the PACT course. Snce t s well known that the soluton of breakage PBEs s problematc because they are partal ntegrodfferental equatons and that soluton technques vary dependng on the models used [5], the use of a computer program s expected to beneft the students. The software s purpose s to facltate nstructon by allowng the students to concentrate on the concepts beng taught rather than on soluton technques, to help make abstract concepts more tangble, and to ntegrate new research developments nto the course. The software wll enable the students to study the effects of the varables, whch would ad the students n makng desgn decsons. At ths tme, the software s lmted to only modelng partcle breakage. A case study usng ths software wll be presented. Breakage Equaton A populaton balance equaton (PBE) s commonly used to account for changes n partcle sze n a collecton of partcles. The partcle sze may be represented by a characterstc length, L, or by a volume, v. The PBE ncludes changes n the partcle sze dstrbuton (PSD) due to varous mechansms ncludng agglomeraton, growth, nucleaton, and breakage of partcles. The number dstrbuton of partcles s typcally represented by the number densty functon, where n(v) s the number densty functon based on the partcle volume. The evoluton of the number densty functon wth tme s gven by the contnuous PBE. A typcal expresson for the breakage equaton s shown n the followng equaton where the frst two terms on the rght hand sde account for the agglomeraton, the next term accounts for growth and nucleaton, and the last term accounts for breakage. 1 Msssspp State Unversty, Box 9595, Msssspp State, MS 39762, phll@che.msstate.edu

2 n(v) t = 1 2 v a(v w, w) n(v w) n(w) dw G V(v) n(v) + v v n(v) a(v, w) n(w) dw b(v, w) S(w) n(w) dw S(v) n(v) (1) The agglomeraton kernel s a(v-w, w) where v and w are partcle volumes; and the growth rate s G(v). In ths paper only partcle breakage s consdered. Ths s easly handled n the PBE by settng the growth and agglomeraton rates to zero. Ths wll be referred to as the breakage equaton snce t only has one mechansm. In the breakage equaton shown below, the frst term on the rght hand sde s the brth term that accounts for partcles larger than v breakng and formng chld partcles n the sze range v + δv, and the last term s the death term. n(v) t = v b(v, w) S(w) n(w) dw S(v) n(v) (2) The death term accounts partcles of sze v that break and fall nto smaller sze ranges. The functon S(v) s the specfc rate of breakage that accounts for the effect of partcle sze on the breakage rate. The brth term ncludes the breakage dstrbuton functon b(v, w) whch s the probablty of a chld partcle of volume v beng formed due a parent partcle of volume w breakng. In the breakage equaton, there are many possble models for the breakage dstrbuton functon. The degree of dffculty n solvng the equaton often depends on the form of the breakage dstrbuton functon. The soluton method uses a dscretzaton technque developed by Hll and Ng [6]. The dscretzed breakage equaton s d dt N = j=+1 β j b j S j N j δ S N where b j s the dscretzed number-based breakage functon, no./no.; N s the number of partcles n nterval, no./ m 3 of slurry or no./ kg of powder; S s the dscretzed volume-based specfc rate of breakage, mn -1 ; and β and δ are the brth and death term factors, respectvely. Ths technque guarantees that mass s conserved durng partcle breakage. The dscretzaton allows easer soluton as a set of ODEs, and allows flexblty n the selecton of the breakage dstrbuton functon. Also, t allows the user to set the parameters for the breakage dstrbuton functons as well as the breakage rate parameters, and the length of tme that breakage occurs. Breakage Dstrbuton Functons The breakage dstrbuton functon b(v, w) s the probablty that a chld partcle of volume v s formed when a parent partcle of volume w breaks. That s, t s the expected dstrbuton of chld partcles for a parent partcle of a gven volume. A varety of emprcal and theoretcal expressons exst for the breakage dstrbuton functon. One emprcal breakage dstrbuton functon s φγ 3w γ ( 1 φ) β -1-1 b (v, w) = ( v / w) 3 + ( v / w) 3 (4) 3w β where φ, γ, and β are constants that depend on the materal beng broken [7]. The product functon wth a power law form has a theoretcal bass [8] s as follows m ' p v (w - v) b v(v, w) = m + (m + 1 w m + (m + 1)(p -2) )(p-1) [ m + (m + 1)(p -1)]! m! [ m + (m + 1)(p -2)]! where p s the number of chld partcles formed and m s a constant selected by the user. (3) (5) IMPLEMENTATION The software for performng the calculatons uses an MS Excel nterface. The nput data s entered nto the spreadsheet nto the cells wth a yellow background as shown n Fgure 1. To run the smulaton, the calculate

3 button s selected. Clckng on the Calculate button causes a vsual basc program to run. Ths vsual basc program uses the nput data from the flowsheet, sends the data to and calls a dll fle that was orgnally wrtten n Fortran, runs the Fortran subroutne, and returns selected results to the Excel spreadsheet. The Fortran subroutne solves the dscretzed PBEs usng a Runga-Kutta method. Input Parameters Austn Parameters Product Law Parameters Ph.36 Rho g/cm 3 Sc 1/mn Gam 2.8 Sc.5 1/mn Alpha Beta 3.5 Alpha p 5 # chldren Delta 2 PrntTme 2 mnutes m exponent PhV MaxTme 1 mnutes L()/L(-1) L(1) 12.5 μm Fgure 1. Example of the data nput secton of spreadsheet. The software also allows the user to enter an ntal sze dstrbuton as well as set the total tme for partcle breakage and tme ntervals where the breakage results are reported. Whle t s well known that usng computer software doesn t automatcally mprove student learnng, t can be used effectvely when there s no other known method for accomplshng the same goal. In ths case, t allows students to solve the PBE n order to demonstrate the effects of the model parameters. CASE STUDIES Students may perform several case studes wth the program. These can consst of studyng the effects of a model s parameters, or nvestgatng the dfferences between models. Case I: Effects of parameters on an emprcal breakage dstrbuton functon One example of a homework problem would be to gve the students an ntal partcle sze dstrbuton before partcle breakage, as well as the breakage rate parameters. In addton, state that the Austn breakage dstrbuton functon, Eq. (4), would be used wth the parameters φ =.36, β = 3.8, and γ = 2.5. Ask the students to run the model for the base case and wth at least three other values of γ. Have the students explan how changng the parameter γ affects the resultng dstrbutons after 3 mnutes of breakage. One possble set of results for ths problem s shown n Fgure 2. The sze of each sze nterval s shown as the major axs length n mcrons. The number fractons for the ntal unbroken partcles are shown for comparson purposes. From examnng Fgure 2, students should recognze that more small partcles are formed as γ s decreased, and that the number fractons of partcles larger than 6 mcrons decrease as γ s decreased. Smlar problems can be assgned where the parameters φ and β are nvestgated. Case II: Effects of parameters on a theoretcal breakage dstrbuton functon Another student exercse would be to nvestgate effect of the parameter p n the breakage dstrbuton functon n Eq. (5). In ths case the ntal partcle sze s gven wth the breakage rate parameters, and the parameter m s set to a value of 2. The student are asked to run the smulaton at a least 4 values of p ncludng p = 3. They are also asked to explan how changng the number of chld partcles p affects the resultng dstrbutons after 3 mnutes of breakage. Fgure 3 s one example of the possble results. The graphs can dffer dependng on the chosen values of p. As n Case I, the sze of each sze nterval s shown as the major axs length n mcrons and the number fractons for the ntal unbroken partcles are shown for comparson purposes. The students should recognze that 1) the breakage dstrbuton functon causes a second peak to be formed between 2 and 3 mcrons, 2) the heght of ths second peak ncreases wth p, and 3) the locaton of the peak shfts to lower partcle szes as p s ncreased. Alternatvely, students could be asked to nvestgate the effects of the parameter m rather than p.

4 Number Fracton Major axs, μm Unbroken gamma = 2.1 gamma = 2.5 gamma = 3.1 gamma = 3.5 Fgure 2. Effect of γ on the resultng sze dstrbuton usng the emprcal breakage dstrbuton functon. Number Fracton Unbroken p = 3 p = 4 p = 5 p = Major axs, μm Fgure 3. Effect of p on the resultng sze dstrbuton usng the theoretcal breakage dstrbuton functon. Comparson of Cases I and II A thrd exercse would be to ask the students to explan the dfferences between the results shown for Cases I and II. The goal would be to help the students understand the dfferences between the two breakage dstrbuton functons. It s expected that the students would realze that the dfferences would partally depend on the parameter settngs for each model. However, the students should also recognze that one dfference between the two breakage

5 functons s that the theoretcal breakage functon produces a secondary peak between 2 and 3 mcrons, whle the emprcal breakage functon has a strong effect on the number fracton of partcles n the smallest sze nterval. Another exercse for helpng students understand the dfferences between the two breakage dstrbuton functons would be to provde them wth an ntal PSD and a fnal PSD and ask them whch breakage dstrbuton functon s best able to descrbe the changes n the PSD. From studyng the resultng PSDs n Cases I and II, they should be able to look at the shape of the resultng PSDs and qualtatvely compare them to the gven data to determne whch functon s a better match. Addtonal Exercses Other exercses are possble to nvestgate other aspects of the models. For example, the students can compare the results at varous resdence tmes to determne the effect of tme on the PSD, nvestgate the effects of breakage rate parameters, and smulate the results of combnng two breakage dstrbuton functons. BENEFITS The benefts to the students and the nstructor are that the program s flexble and that other exercses can be developed to demonstrate breakage concepts. The advantages to the students are that they 1) can nvestgate the effects of the model parameters wthout beng encumbered wth a soluton technque, 2) do not need to know a programmng language to use the program, 3) can study two dfferent breakage functons ndependently, and 4) can study the combnaton of two dfferent breakage dstrbuton functons. CONCLUDING REMARKS The development of ths program allows students to study the effects of varous parameters on two dfferent breakage dstrbuton functons. Varous case studes can be performed to demonstrate the effects of varous parameters as well as the nteractons between parameters. In addton, t allows the students to compare and/or combne two dfferent breakage dstrbuton functons. It also provdes a tool for generatng smulaton results to compare wth expermental data. Future work ncludes expandng the program to more breakage models, and creatng programs wth other mechansms such as agglomeraton. It should be noted that many breakage dstrbutons exst n the lterature and there s no plan to nclude every model n ths tool, but to provde the nstructor and the student wth some representatve examples. Ths software has the flexblty to allow students to run smulatons that demonstrate concepts n partcle breakage, but avods the dffcultes of solvng the equatons. ACKNOWLEDGMENT Ths materal s based upon work supported by the Natonal Scence Foundaton under Grant No REFERENCES [1] Hale, J. M., Toward techncal understandng. Part 1. Bran structure and functon, Chem. Engr. Educ., 31, 1997, [2] Felder, R. M., and R. Brent, FAQs, Chem. Engr. Educ., 33, 1999, [3] Felder, R. M., D. R. Woods, J. E. Stce, and A. Rugarca, The future of engneerng educaton. II. Teachng methods that work, Chem. Engr. Educ., 34, 2, [4] Wankat, P. C., The Effectve, Effcent Professor: Teachng, Scholarshp, and Servce, Allyn and Bacon, Boston, 22, [5] Ramkrshna, D., Populaton Balances: Theory and Applcatons to Partculate Systems n Engneerng, Academc Press, New York, 2, [6] P. J. Hll and K. M. Ng, "New dscretzaton procedure for the breakage equaton", AIChE J., 41, 1995,

6 [7] Austn, L., K. Shoj, V. Bhata, V. Jndal, K. Savage, and R. Klmpel, Some results on the descrptonof sze reducton as a rate process n varous mlls, Ind. Eng. Chem., Process Des. Dev., 15, 1976, [8] P. J. Hll and K. M. Ng, "Statstcs of multple-partcle breakage", AIChE J., 42, 1996, Prsclla J. Hll Prsclla Hll s currently an Assocate Professor n the Dave C. Swalm School of Chemcal Engneerng at Msssspp State Unversty. She receved her B.S. and M.S. degrees n chemcal engneerng from Clemson Unversty; and her Ph.D. n chemcal engneerng from the Unversty of Massachusetts at Amherst. She has research nterests n crystallzaton, partcle technology, populaton balance modelng, and process synthess.