Transcription

1 Ice crytal nucleaton, growth and breakage modellng n a craped urface heat exchanger H. Benkhelfa (a, M. Arellano (a,b G. Alvarez (b, D. Flck (a (a UMR n 45 AgroParTech/INRA, Ingénere-Procédé-Alment, 6 rue Claude Bernard, 752 Par Cedex 5, France (benkhel@agropartech.fr, flck@agropartech.fr (b Cemagref UR Géne de Procédé Frgorfque. BP 44, 926 Antony, France (marcela.arellano@cemagref.fr, gracela.alvarez@cemagref.fr ABSTRACT The objectve of th tudy to propoe a mathematcal model of ce crytallaton occurrng durng the freezng of ucroe oluton n a contnuou freezer. The approach developed a -D radal model where ce crytal nucleaton, growth and breakage knetc are decrbed by dcrete populaton balance (PB equaton coupled wth computatonal flud dynamc (CFD modelng of the energy conervaton equaton. In th approach, the problem ha three dmenon: one crytal ze varable, one poton varable, correpondng to the radu between the craper and the wall and the redence tme varable. For the numercal mplementaton, the populaton balance equaton dcretzed by ung the cla method. Populaton denty then obtaned for each cla of crytal ze. Ice fracton and orbet temperature are obtaned n functon of redence tme and exchanger radal coordnate. Aumng phercal partcle, the mean crytal dameter calculated. The reult are compared to expermental meaurement of temperature and mean crytal ze at the freezer outlet for gven proce condton: craper rotatonal peed, flow rate and wall evaporator temperature. Wth a frt etmated et of model parameter, t hown that the expermental tendence are correctly repreented by the model. Th coupled CFD-PB modellng approach can then be condered a a promng tool for the undertandng and the predcton of ce crytallaton for everal operatng condton n a contnuou freezer. Keyword: Ice crytallaton; ucroe oluton, craped urface heat exchanger; computatonal flud dynamc; populaton balance equaton. INTRODUCTION Scraped urface heat exchanger (SSHE are currently ued for the producton of ce lurre, orbet or ce cream [], but alo for frut juce concentraton. The product qualty manly governed by the ce content and crytal ze dtrbuton whch are dependent on how the crytallaton occur n the SSHE. In order to aure a contant product qualty wth mnmum energy conumpton or to mprove the qualty, manufacturng procee can be modelled and optmzed. An adequate modellng of th proce mple the undertandng of flow behavour wthn the SSHE, t effect on heat tranfer and alo the knetc of crytallaton nvolved n the ytem. But there are very lmted tude on the nfluence of proce parameter on crytallaton knetc durng freezng, partcularly when ce concentraton hgh uch a n orbet or ce cream at the SSHE outlet []. In th cae, the crytal ze ha a great nfluence on the product rheology and on t qualty. It then neceary to take nto account the crytal nucleaton, growth and breakage to correctly model and etmate the product crytal ze. Th knd of model lead to populaton balance equaton (PB and are often ued for ndutral crytallaton procee whch are aumed a perfectly mxed o that homogeneou temperature and concentraton are condered: th the well known mxed upenon mxed product removal model (MSMPR [2]. Thee aumpton can not be made n the cae of a SSHE. Therefore n a SSHE model, temperature, compoton of lqud phae and crytal ze dtrbuton depend on poton. A combned CFD and populaton balance modellng approach have then been ued. The combned CFD-PB modellng ha recently been ued n 2-D for the mulaton of paracetamol crytallaton n an agtated veel [] and of water crytallaton n a craped urface freezer [4]. In both cae, the flud flow modeled at teady tate. The man lmtaton of the lat tudy are the phycal mplfcaton (vcoty of the orbet equal to the ntal one and the nuffcent numercal dcretzaton of the PB equaton. The objectve of our tudy to propoe a mathematcal model of ce crytallaton occurrng durng the freezng of ucroe oluton n a contnuou freezer, by takng nto account crytal nucleaton, growth and breakage phenomena, wth a uffcently hgh numercal dcretzaton of the PB equaton and for everal

2 proce condton. The coupled CFD-PB approach ued to determne the populaton denty, crytal ze, ce fracton and orbet temperature n functon of redence tme and exchanger radal coordnate. The reult are compared to expermental meaurement of temperature and mean crytal ze at the freezer outlet for gven proce condton: craper rotatonal peed, flow rate and wall evaporator temperature. MATERIALS & METHODS Modellng approach Th approach developed at the cale of a repreentatve elementary volume (REV correpondng to a mall volume compared to the macrocopc cale but contanng a large number of ce crytal. Crytal of crtcal ze appear (nucleaton n th REV and grow n functon of the devaton to thermodynamc equlbrum. The devaton to thermodynamc equlbrum, alo called under-coolng, can be characterzed by the dfference between local temperature and local freezng pont whch depend on olute concentraton n the oluton. The ce crytal can break dependng on the craper rotatonal peed, the ce content and ze dtrbuton. The problem tuded wth a -D radal approach and ha three dmenon: one crytal ze varable, one poton varable, radu between craper (R and wall (R e and the redence tme varable (Fgure. t t t L + L L L c R r r+ r R e Mxng Growth dffuon Breakage Homogeneou nucleaton: Heterogenou nucleaton: Fgure. Schematc repreentaton of crytal nucleaton, growth and breakage n a SSHE In our cae, n order to lmt computng tme and memory, the geometrcal decrpton of the crapng blade replaced by an effectve radal dffuvty D m. Th mxng dffuvty repreent the mxng edde n the crapng blade zone (Fgure. The ce cream crytallzer aumed to behave a a plug flow reactor. The populaton balance of the crytal ze dtrbuton conder crytal growth, nucleaton and breakage a well a the radal dffuon. The populaton balance equaton then wrtten a follow: ψ ψ + (G. ψ = N δ( L Lc δ( r Re + Bb ( L + ( rdm ( t L r r r wth G and N the growth and nucleaton rate. δ the Drac functon meanng that nucleaton create crytal wth a ze equal to L c and that only heterogeneou nucleaton at the freezer wall (r=r e condered here. G can be negatve n the cae of crytal meltng n warm temperature zone. The nucleaton and growth rate depend on the degree of under-coolng and can be expreed n the followng form: 2 πr (2 e N = α. ( T δ at Te V G = β (T at T γ ( where T at et T e are repectvely the aturaton temperature and the temperature at the external wall (r=r e (whch cloe to the refrgerant evaporaton temperature, α and β are knetc parameter and V the freezer volume. B b (L repreent the net ncreae of partcle number by breakage. The breakage a functon of the craper peed N crap, of the crytal ze L, the populaton denty ψ(l and the ce fracton ϕ. It aumed that a crytal break nto two maller crytal havng half of the ntal volume, the net ncreae of partcle can therefore be expreed a: ν ν B (L = 2εΝ. 2 L. ψ ( 2L. ϕ εn.l. ψ ( L ϕ ( b crap crap

3 Wth ε the breakage coeffcent. In th approach, the heat tranfer balance equaton can not be wrtten excluvely n functon of temperature. Indeed, the orbet nternal energy depend on ce fracton whch not an explct functon of temperature. The energy balance equaton then expreed wth the volumetrc nternal energy, u, a tate varable. 2 u T u = ( rλapp + Dm + µ γ t r r r r (4 Wth u = ρ H fϕ + ρ ( ω C + ( ω C w T. H f the pecfc fuon latent heat, C and C w are olute and water pecfc heat capacte, ρ the denty of ce, ρ the denty of oluton and ω the ntal ucroe ma fracton. The vcou dpaton expreed n functon of an effectve hear rate: γ = χ 2 π N crap Re R R wth χ the vcou dpaton coeffcent. Conderng crytal a phercal partcle, the volume ce fracton ϕ can be expreed at poton r and tme t a: ϕ ( r,t = or Lmax or Lc πl ψ ( L,r,t dl 6 The energy balance equaton (Equaton 4 become an Integro-Dfferental equaton and completely coupled wth the populaton balance equaton (Equaton. The boundary condton are: at r=re=25mm: ψ/ r =, T=T e at r=r=6mm: ψ/ r =, u r =. The populaton balance equaton dcretzed ung N clae of L ze, wth the frt cla centred around L c a hown on Fgure. An upwnd cheme ha been choen for the growth (f (T-T at <, crytal are growng o G> or ele crytal are meltng and G<. Thee N dfferental PB equaton are completed wth the energy conervaton equaton (Equaton 4. Thee N+ equaton were olved by ung the Fnte Volume Element Method wth a pecfc Matlab code. The reult were found to be ndependent of the number crytal ze clae for of N 2. Populaton denty ψ(r,t then obtaned for each cla. Ice fracton ϕ and orbet temperature T are obtaned n functon of redence tme and exchanger radal coordnate r. N N Aumng phercal partcle, the mean crytal dameter calculated wth: d = ψ ψ L Expermental meaurement The expermental craped urface heat exchanger a WCB MF5 plot cale freezer. Th equpment ha a nomnal output from 25 to kg/h. The nner dameter of the heat exchange tube,5m and the length,4m. Inde th cylnder, a old equpped wth two craper blade. The refrgerant flud R22 (Chlorodfluoromethane contnually vaporzng (between - C to -25 C wthn the veel jacket. The daher rotaton peed can be vared from to rpm. Sorbet mx ntroduced nto the ytem by a pton pump under potve preure. The draw temperature of orbet meaured by a calbrated Pt probe. The probe nerted nto the orbet at the outlet ppe of the freezer. The accuracy of th enor. C. The ce crytal ze dtrbuton of orbet ha been meaured on-lne by the Focued Beam Reflectance Method (FBRM. The FBRM probe (Model S4A-8 manufactured by Laentec, Mettler Toledo, USA. The ue of th probe durng orbet crytallaton wa decrbed n a prevou paper [5]. The crytal number and chord length are meaured contnuouly. RESULTS & DISCUSSION The above model to be complemented wth conttutve equaton for varou product properte. e (5

4 Conttutve equaton The ma fracton of olute n the unfrozen phae gven by: ρ ω = ω ( φ ρ For the orbet apparent conductvty, the followng Maxwell relaton gven n the cae of a upenon of a old phae n a contnuou lqud phae [6]. 2λ ( ( + λ 2φ λ λ λapp = λ 2λ + λ + φ λ λ wth: λ = ,975 - (-T.56 for the ce (λ n W.m -.K - and T n C and λ = (-ω(,56+,976 - T -7,847-6 T² + ω(,266+,8 - T -2,8-6 T² for ucroe oluton. The vcoty of the ce lurry modelled a follow: µ=µ (+ 2.5 φ +.5 φ 2 +ξ.27 exp(.6 φ whch a modfed Thoma equaton, wth an adjutable parameter ξ.the Thoma equaton generalze the reult of Enten law for hgher old concentraton. µ the vcoty of the contnuou phae of ucroe n water oluton and gven by ( P. 4 M. 4. P µ + + = wth P = ω * /( 9 8ω and M = ( T /( 9+ T Etmaton of model parameter The model parameter are recalled on Table. Some of the parameter are determned by conderng drect expermental meaurement or wth prevou work publhed n lterature. Some of them are etmated by fttng the model wth output meaurement of temperature and mean crytal chord allowed by the on-lne enor. The mean ce crytal dameter obtaned by the model tranformed nto a mean ce crytal chord correpondng to the on-lne meaurement. Th mean chord equal the mean ce crytal ze multpled by p/4: Table. Model parameter and etmated value Decrpton Model parameter Range Unt Wall heat tranfer coeffcent h e 2 W.m -.K - Nucleaton contant α m -2 K - Nucleaton exponent δ 2 - Growth contant β m. -.K - Growth exponent γ - Breakage contant ε 2 m - Vcou dpaton coeffcent χ 2 - Vcoty parameter ξ 5 - Intal crytal ze L c 5 µm Péclet number Pe 64 - A frt etmaton of the wall heat tranfer coeffcent how that th value cloe to 2 W.m -.K -. The growth and nucleaton exponent were choen repectvely equal to and 2 a other author dd [4]. In the ame way, the growth and nucleaton contant are cloe to the one obtaned by Lan [4] n the cae of freezng of ucroe oluton. The value are of ame order of magntude but reman dfferent becaue of the proce and of the product that are dfferent. They are then ftted wth the model.

5 The mxng dffuvty can be expreed by ung a Péclet number Pe wth a tangental velocty defned at the craper blade tp and a mxng length equal to the radal dtance wept by the craper (R e -R. vcrap( Re R Dm = Pe An expermental dentfcaton howed that n our cae the Péclet number can be condered a contant, Pe=64 [7]. Fgure 2 how the evoluton of everal varable along the longtudnal dmenon of the freezer (for the et of parameter of table. The value dplayed at each pont the urface average (n the radal drecton of the repectve quantty T T Tat..25 Ice fracton φ ph g T at nlet outlet x Mean chord Cmean C mean mu Vcoty µ mu Fgure 2. Model output of everal varable: axal profle of product temperature and equlbrum temperature (T at, ce fracton, mean chord and vcoty.( for a craper peed of 75 rpm, an evaporator temperature of -5 C and a mx flow rate of 49.8 kg.h - Th fgure 2 how that the ce fracton, the mean chord and the product vcoty ncreae when the product temperature below the equlbrum temperature correpondng to the begnnng of crytallaton due to crytal nucleaton and growth. Comparon between experment and model reult A comparon between expermental and model reult made for the ame proce condton. Fgure how the nfluence of evaporaton temperature on mean chord length at the freezer output for a gven craper peed and mx flow rate. The nfluence of evaporaton temperature on the product output temperature for a gven craper peed and mx flow rate howed on Fgure 4. The expermental reult how that the lower the evaporaton temperature the lower the output temperature and the mean chord length. Th phenomenon well know n lterature and can be explaned by a hgher freezng peed leadng to more nucleaton mechanm. Low evaporaton temperature generate a large amount of mall nucle wth le ndvdual growth. The model reproduce th phenomenon gvng the ame behavour for the mean chord length. A relatve good agreement between expermental and mulated reult oberved. The frt etmated parameter can then be conderer a content. For thee condton, the mean devaton between experment and model equal 6%.

6 C out (µm Experment 4 Modele T evap ( C Fgure. Influence of the evaporaton temperature on the mean crytal chord for expermental and model reult for a craper peed of 75 rpm and a mx flow rate of 49.8 kg.h -. Tout ( C T evap ( C Experment Fgure 4. Influence of the evaporaton temperature on the product output temperature for expermental and model reult for a craper peed of 75 rpm and a mx flow rate of 49.8 kg.h -. Modele CONCLUSION Th paper preent a coupled CFD-PB approach to model ce crytallaton n ucroe oluton occurrng n a contnuou freezer. The populaton denty, crytal ze, ce fracton and orbet temperature are determned n functon of redence tme and exchanger radal coordnate. A frt et of etmated parameter ued to reproduce the man expected behavour. The reult are compared to expermental meaurement of temperature and mean crytal chord at the freezer outlet for gven proce condton. Expermental reult and model calculaton compare reaonable and the model parameter eem content. Th model wll be ued n reduced form a a predctve tool n the cae of the control of ce freezng proce n the European Computer-Aded Food procee for control Engneerng project (CAFE n KBBE n the European Communty' Seventh Framework Programme (FP7/27-2. REFERENCES [] Hartel R. W. (996. Ice crytallzaton durng the manufacturng of ce cream. Trend n Food Scence & Technology [2] Randolph A.D., Laron M.A. (988. Theory of partculate procee. Acad. Pre. New York. [] Woo X.Y., Tan R.B.H., Chow P.S., Bratz R.D. (26. Smulaton of mxng effect n antolvent crytallzaton ung a coupled CFD-PDF-PBE approach. Crytal Growth &Degn, 6, 29-. [4] Lan G., Moore S. and Heeney L. (26. Populaton balance and computatonal flud dynamc modellng of ce crytallaton n a craped urface freezer. Chemcal Engneerng Scence, 6, [5] Haddad A., Benkhelfa H., Alvarez G., Flck D. (2. Study of crytal ze evoluton by focued beam reflectance meaurement durng the freezng of ucroe/water oluton n a craped urface heat exchanger. Proce Bochemtry. 45 ( [6] Kavany M. (995. Prncple of heat tranfer n porou meda. 2nd ed. Sprnger-Verlag New York. [7] Haddad Amamou A. (29. Couplage entre écoulement, tranfert thermque et tranformaton lor du changement de phae d un produt almentare lqude complexe Applcaton à la maître de la texture. Ph-D The. AgroParTech. Par. France.