Modeling the Filament Formation in the Wet Spinning of Synthetic Fiber from Polymer Solutions

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1 heoecal Foundaons of hemcal Engneeng Vol. 3. No pp anslaed fom eoecheske Osnovy Khmchesko ekhnolog. Vol. 3. No pp Ognal Russan ex opygh 1996 by Kalabn Paksve Kukushkn. Modelng he Flamen Fomaon n he We Spnnng of Synhec Fbe fom Polyme Soluons A. L. Kalabn E. A. Pakshve and N. A. Kukushkn ve Sae echncal Unvesy ve Russa All-Russa Reseach Insue of Synhec Fbe Moscow Russa Receved Ocobe Absac-A mahemacal model s poposed fo gelaon ha akes place as he coagulan dffuses no polyme soluon flamens. Gelaon knecs s modeled numecally. he manne n whch he sze of he complee gelaon zone vaes wh me s deemned and analyzed. A compason s made beween expemen and calculaon. Dawng upon he model he dependence of gelaon me on basc pocess vaables s esmaed. INRODUION Bascally he we spnnng of synhec fbe ncludes pumpng he sang polyme soluon hough spnnees and hen passng he flamens hus poduced hough a coagulang bah. As he polyme soluon passes hough he bah he coagulan fnds s way no he polyme flamens and accumulaes hee. When enough of he coagulan has accumulaed he polyme soluon s sepaaed no a sold gel-lke and a lqud phase. he gel-lke phase s he pmay sucue of he fbe and govens s popees n many especs. Opmzaon and conol of he spnnng pocess especally when vaous ypes of fbe need o be spun ae complcaed opeaons whch eque a gea effo and hghly sklled pesonnel o pefom. o enhance he elably and speed of he acon s essenal o have an adequae model of gelaon ha s of he fs sage n fbe spnnng [1]. Many nvesgaos have aemped o develop such models [1-7]. Unfounaely all of he models fal o ake no accoun some mpoan aspecs of he sucung pocess such as he phase equlbum of he polyme-solven-coagulan sysem he empeaue behavo of he knecs he knecs of phase sepaaon elaxaon pocesses ec. Exsng models of we spnnng [1-7] fal o esmae he knecs of sucung ha s o esablsh he paen of changes n he hckness of he soldfed polyme wh me. R g of he gelaon zone changes wh changes n pocess vaables. A model ha consdes he knecs of gelaon s poposed n

2 [8] bu s based on fa oo many assumpons. he am of hs pape s o develop a model fo he nal sage of fbe spnnng flamen fomaon fom a polyme soluon by he we spnnng pocess usng exsng models and akng no accoun he phase equlbum of he polyme-solven-coagulan sysem hea ansfe he evoluon of dffuson pocesses wh empeaue he exsence of a layeed wo-phase sucue when flamens ae fomed fom a polyme soluon he hemal effecs assocaed wh he eacons of he componens nvolved n he pocess and changes n fbe sze n he pocess. HE MODEL In developng he model we make he followng assumpons abou flamen fomaon. 1 he poblem s eaed unde axal symmey n a one-dmensonal saemen assumng ha he coagulan dffuses solely along he adus n he absence of convecon and ha he fbe s an unbounded cylnde whose adus s small n compason wh he lengh of he coagulang bah. 2 Only he dffuson of he coagulan and solven s consdeed assumng ha he polyme does no dffuse no he coagulan. 3 he dffusvy of he coagulan no he polyme soluon D 1 dffes fom s dffusvy no he soldfed gel-lke laye D beng n he ao D D 1 /3 [1] and boh ae funcons of empeaue.. 4 Mass ansfe occus unde bounday condons of he fs ype assumng ha dung he complee gelaon me he coagulan concenaon n he bah a he fbe changes no moe han 3% [2] a a soluon flamen velocy of abou.1 m/s. 5 Dung gelaon he gel laye shnks unfomly acoss s ene hckness. 6 he hemophyscal coeffcens of he polyme soluon and of he soldfed laye ae funcons of he empeaue and ae ndependen of he coagulan concenaon ; hea ansfe a he fbe suface obeys Newon's law ha s hd-knd bounday condons. 7 We neglec he change n he flamen adus caused by mass ansfe beween he flamen and he coagulang bah and by he exenson of he flamen. 8 We consde he fomaon of only one flamen n he coagulang bah and neglec any neacon beween flamens. Keepng n mnd he aonale of he we spnnng pocess and he daa fom [1-8] we wll consde he dffuson of he coagulan no he fbe and of he coagulan fom he fbe no he coagulang bah. he applcable equaon of dffuson n cylndcal coodnaes akes he fom С 1 [ D ] 1 hs s a sysem of fou mass ansfe equaons ha descbe he dffuson of he coagulan no he polyme soluon fo and no he gel fo 1 and he especve dffuson of he solven fo 2 and 3. Hea ansfe n he fbe s descbed by he equaons

3 v j j q a + ] [ 1 2 hs s a sysem of wo equaons fo j n he case of he polyme soluon and fo j 1 n he case of he gel. he nal condons a ae j nj.. n. 3 he bounday condons ae as follows. A he oue bounday of he fbe ha s R ' ] [ R s R R α λ 4 R s 2 a he polyme soluon-gel neface ha s R d D D R λ λ 6 1 R R A he cene of he fbe ha s. he symmey condons ake he fom he hea souce n equaon 2 accouns fo he hea eleased upon mxng he coagulan and he solven [12] and s powe s G q q m v 8

4 Gelaon causes a change n he volume [3] and geomecal dmensons of polyme soluon flamens. he change comes abou because he gel akes up a smalle volume han does he polyme soluon. he fbe hckness s a vaable R o R and can be found fom he equaon R 2 R 2 +γ [R 2 -R 2 ]. 9 he pesence of a movable bounday poses addonal dffcules n solvng he poblem. One way o change fom a no saonay o a saonay bounday s o use he von Mses ansfomaon [9] whch noduces a new dmensonless vaable defned as /R. he change fom he ognal ndependen vaables and o he new ones ŋ and s done usng he expessons 1 R 1 A Upon he change of vaables and afe some algebac manpulaons equaons 1 hough 8 ake he fom + A D R ] [ 1 2 1a v j j q A a R + + ] [ a j nj n 3a Fo 1

5 λ 1 α[ 1 s ] R 4a 1 s 2 Fo R/R + 1 D D + 1 5a +1 2 λ λ 1 1 6a 1 Fo 1 7a qm qv A 8a G In wha follows we wll use he ognal ndependen vaables and o make he pesenaon moe nsucve and he new vaables and wll be used n solvng equaons 1a hough 8a. In calculang changes n he gelaon fon wh me R s necessay o consde he phase sepaaon dagam of he hee-componen polyme-solven-coagulan sysem. I esablshes he condons fo phase sepaaon gelaon. he gel hckness s found fom he anscendenal equaon

6 [R] c [] 11 whee c s found fom he phase dagam. hs equaon s solved fo he gelaon fon coodnae R R. hen he pesen gel hckness R g can be defned as R g R R. Usng hs elaon and equaon 11 we ae n a poson o fnd he gel hckness as a me funcon R g. We solve he nonlnea sysem of equaons 1 hough 11 usng an mplc fne-dffeence scheme [9 1]. he von Mses ansfomaon does no complcae he ask. he algebac sysem of equaons s lneazed by an eave mehod [1] whch akes no moe han hee eaons seps. Ieaons ae used because he behavo of R o s no known n advance and has o be deemned n he couse of he soluon. o see f s coec he numecal soluon s checked agans known soluons o lnea poblems fom [11] and n he case of nonlnea poblems agans a check example fom [1]. Wh he chosen paamees of he numecal scheme he eo s no geae han 1.5%. Fo numecal smulaon we choose he sof ype of we spnnng usng he polyacylonle PAN- dme-hyl fomamde DMF-wae sysem. he nal concenaons n he soluon ae 2% PAN and 8% DMF and n he coagulang bah 8% DMF and 2% wae. he pocess vaables ae R.5 mm a n 5 and s -1. he hea and mass ansfe coeffcens ae chosen as follows. Accodng o [3] he dffusvy of he coagulan no he soluon vaes wh empeaue so we choose o be D x 1O -1 m 2 /s and ha of he solven no he soluon D 3 14D 1.he dffusvy no he soldfed polyme s assumed o be D D +1 /3.2 afe [1]. he coeffcens necessay o solve he hemal poblem nvolvng he PAN-DMF soluon ae boowed fom [12] neglecng he empeaue dependence: λ.2 W/m K G 21 J/kg K and ρ 95 kg/m 3. Fo lack of daa on he hemal popees of he gel we assume hem o be he same as fo he polyme soluon. he value of α 1 W/m 2 K s deemned afe [1]. he conbuon due o he hea of mxng equaon 8 o he hea defned n equaon 2 s esmaed o be such ha can cause a change n empeaue no moe han.5% of he flamen empeaue ange fo he sysem n queson and unde he spnnng condons chosen gven he same ode of magnude fo he me devaves of empeaue concenaon and specfc hea of mxng q m of wae and DMF [13]. he knecs of gelaon s calculaed usng he phase dagam fo he PAN-DMF-wae sysem fom [14]. A gaph elang he hckness of he soldfed polyme gel o me R g s shown n Fg. 1. hs s an 5-shaped cuve mplyng ha a he sa of he pocess he hckness R g s popoonal o he squae

7 oo of Fg. 1. Gel hckness as a funcon of me R g. 1 Fo R ; 2 fo R o cons; 3 flamen hckness as a funcon of me R. Fg. 2. Fbe empeaue aveaged ove adus as a funcon of me. me 1/2 hen uns almos lnea and. as me passes becomes popoonal o a hghe powe of me. In he subsequen calculaons we assume ha R R cons. hs assumpon does no cause a qualave change n he behavo of he elaons gven below bu makes hem moe llusave. We calculae how he empeaue of he fbe vaes wh me. he esuls ae gven n Fg. 2. Noe ha he fbe cools o amben empeaue n.4 s. whch s abou one-quae of he gelaon me. Fgue 3 llusaes he manne n whch he coagulan concenaon n he fbe vaes wh coodnae and Fg. 4 wh me. he knks n he cuves eflec he fac ha a me he gelaon fon s posoned a a pon whose coodnae s. he knks ae aceable o dsconnues n he

8 concenaon gadens and o he consevaon of coagulan mass fluxes bounday Fg. 3. Dsbuon of coagulan concenaon along adus a dffeen nsans.: 1.3; 2.6; 3.9; 4 1.2;5 1.5 s. Fg. 4. oagulan concenaon n fbe ploed agans me a pons wh coodnaes. 1 ; 2.1; 3.2; 4.3 m. Fg. 5. Gel hckness ploed agans me. R g.1 hough 3 calculaon; 4 hough 6 expemen [2]; 1 and 4 fo R.7 mm: 2 and 5 R { -.15 mm; 3 and 6 R 15 mm

9 condons of he fouh knd equaon 5]. o llusae he fac a pon s hghlghed n Fgs. 1 3 and 4 whch ells us ha a me.3 s he soldfcaon fon s posoned a a coodnae R.3 mm whch coesponds o a gel hckness R g.2 mm. o see f he poposed mahemacal model s wokable and gves a close f o he acual suaon we compae he calculaed esuls wh he expemenal daa fom Mezhov [15] fo PAN fbes we-spun fom an aqueous hodande polyme soluon. Mezhov [15] epos hs obsevaons of changes n he gelaon adus wh me R g fo dffeen spnnee ad. In hs expemens he flamen adus was R and.7 mm. he nal empeaue was equal o ha of he coagulan n he bah sohemal condons n s 11 he coagulan wae concenaon n he bah was 9% and he nal polyme soluon concenaon was n 42.68%. All he wae fee and bound was aken o be he coagulan. In numecal smulaon based on he poposed model we use he phase dagam fo he aqueous hodande soluon [14] Fo calculaons we need o know he dffusvy of he coagulan no he polyme soluon D 1 and he dffusvy of he coagulan no he polyme gel D. We boow hem fom expemenal daa epoed n [15]. Fo a flamen adus R.15 mm he numecally fed dffusvy s D 2.2 x 1-1 m 2 /s fo a ao D D 1 /3. hs value assues a close f beween he expemenal and calculaed complee gelaon me g s. Usng hs value of dffusvy we hen calculae he manne n whch he adus of he soldfed polyme vaes wh me R g fo flamens dffeng n R. he calculaed esuls ae pesened n Fg. 5. A good f wh he expemen can be seen fo wo cuves and a sasfacoy f fo a flamen of adus R.15 mm. Noe also ageemen n he complee gelaon me n he qualave behavo of he 5-shaped cuves and n he me vaaons of R g. he dscepancy beween he calculaed and he expemenal daa may have asen because he mahemacal model s developed unde seveal assumpons and because hee may be some naccuaces n he phase dagam used n he calculaons [14]. he mos cucal pon n he we spnnng of synhec fbes s he buldup of he polyme shell gel ove me. heefoe ou nex ask s o analyze he paen of changes n he gelaon zone sze wh me R g and o explan why hs funcon s S-shaped. o hs end we consde an appoxmae lnea model fo gelaon expessed n he fom of algebac equaons. In dong so we assume ha a smplfed nonlnea model wll peseve he qualave fom of he funcon R g. hs assumpon s based on he followng fac. he nonlneay whch makes possble o calculae he knecs of gelaon causes he dffusvy o change wh me and space only quanavely. heefoe we ae n a poson o ake no accoun he changes ha occu n he dffusvy of he soluon and he gel as he gel egon undegoes dynamc changes. he gel hckness s found fom he anscendenal equaon 11. he poposed gelaon model wll

10 ansfom no he lnea equaon of dffuson fo an unbounded cylnde on he povso ha 1 he spnnng opeaon goes on unde sohemal condons; ha s he nal empeaue of he polyme soluon s he same as ha of he coagulang bah so ha he flamen empeaue emans consan cons; 2 he coagulan has he same dffusvy no he polyme soluon and no he gel; 3 he coagulan has a consan dffusvy. If hese condons ae sasfed equaon 11 s smplfed o he fom [R ] c cons. 12 A smla appoach o he lnea equaon of dffuson s poposed n [6]. Fo compason consde solvng equaons 11 and 12 fo he '"sof" spnnng opeaon n he case of he PAN-DMF-wae sysem. o faclae he compason we assume ha he spnnng vaables have he same values as hey had n he pevous case wh mno changes necessaed by he fac ha he opeaon goes on sohemally he nal empeaue of he soluon s he same as ha of he bah n s -1. In he ccumsances he concenaon coespondng o polyme gelaon n he phase dagam a he specfed empeaue s c 2% afe [14]. We ake ha he dffusvy of he coagulan no he fbe aveaged ove empeaue and me s D 2.8 x 1-1 m 2 /s. In he case of a lnea model he oal gelaon me can be found usng he abulaed soluon of he lnea poblem a he cene of a cylnde gven n [11]. o demonsae he gelaon may be aken o have eached compleon when a he pon mos dsan fom he flamen suface whch s he fbe cene he coagulan concenaon s suffcen o bng abou he phase anson of he polyme soluon o a gel o when c. Fo hs value of c. he oal gelaon me s g 1.3 s. hs esmae s as much as one-hd smalle han he calculaed me g 1.5 s see Fg. 4. he eason s ha a lnea model s unable o ake no accoun he changes ha occu n he dffusvy upon gelaon. he soluon o equaon 12 on he me neval [ g ] s epesened by cuve 1 n Fg. 5. In he same fgue cuve 2 gves he soluon obaned usng he gelaon model. ompason of he soldfed polyme hckness as a funcon of me found fom he model equaons 1 hough 11 wh he soluon of a smla lnea equaon of dffuson 1 shows ha boh behave smlaly n qualave ems. heefoe n he subsequen dscusson we ake ha he esuls obaned fom an analyss of he funcon R g fo he lnea poblem qualavely explan he knecs of gelaon n he case of he nonlnea model as well. Reso o a lnea model does no sgnfcanly smplfy equaon 12- emans anscendenal. heefoe n ode o ansfom equaon 12 o an algebac equaon and o fuhe analyze he behavo of gel hckness wh me we use a well-known mehod fo analyss of soluons o he paabolc equaon of anspo fom [16]. I s based on an appoxmae soluon of a gven anspo pocess a sho and long mes. In hs mehod he gelaon me g s dvded no wo pas. he pon of dvson s he value

11 of he Foue numbe Fo n D n /R 2 o.81 coespondng o he nsan when changes begn n he concenaon a he cylnde's cene ha s he pon fahes fom he flamen's suface o he coagulan souce. o demonsae fo D he value of Fo n coesponds o n.97 s. hs s he nsan whee he concenaon a he fbe's cene s 1 % whch checks wh he assumpons made. In [16] s poposed ha n he fs half-neval whee < n he concenaon should be calculaed usng an appoxmae expesson whch s an exac one fo a sem-nfne medum [1116]. In he second halfneval whee > n hs should be done usng a dffeen appoxmae soluon whch s he fs em of an nfne sees soluons o he lnea equaon of dffuson fo an nfne cylnde [1116]. We can hen oban expessons fo he polyme gelaon hckness n explc fom as a funcon of me fo a cean ccal concenaon c as R g D/[ef -1 { θ }] 1/2 fo < < n 13 whee ef -1 s he nvese of he eo funcon ef θ c s / n - s and R g {R {1-[1- { θ }exp5.79fo-fo 1 ] 1/1.51 } 14 fo g > > n.. he esuls calculaed by equaons 13 and 14 ae gven n Fg. 6. hey agee closely wh he soluon of he nonlnea poblem fo < n and sasfacoly fo > n. I s noewohy ha boh calculaed cuves ae n full qualave ageemen wh he funcon R g deved eale and explan s S-shape n qualave ems. Equaons 13 and 14 mply ha. g R 2 o [Fo 1 -lnθ/5.79]/d n + c 15 whee θ anges beween 1 and. heefoe s obvous ha lnθ <. Equaon 15 may be wen as a sum n whch he fs em n s he me equed fo he coagulan concenaon a he cene of a flamen o change by abou 1% elave o he nal value. In ohe wods he coagulan seam has eached he flamen's cene o he pon fahes fom he fbe's suface. he me c s he me dung whch he coagulan concenaon changes fom s ognal value o he value equed fo gelaon c a a gven empeaue. Wh such an appoach he gelaon pocess may

12 Fg. 6. Gel hckness ploed agans me. Rg: l afe 11; 2 by equaon 12: 3 by equaon 13; 4 by equaon 14. Fg. 7. omplee gelaon me g ploed agans fbe-spnnng vaables: 1 coagulan concenaon n bah s : 2 coagulan concenaon n polyme soluon n 3 bah empeaue s ; 4 polyme soluon flamen adus R o. be hough of as poceedng n wo sages whose duaon s n and c especvely. I s of paccal nees o analyze he way he gelaon chaacescs behave wh changes n he we spnnng pocess vaables. he basc vaable ha chaacezes he soldfcaon of he polyme s he complee gelaon me g Gven a consan fbe velocy v he me g deemnes he bah lengh Lv g. 16 Fgue 7 gves gaphs of he complee gelaon me g ploed agans he we-spnnng pocess

13 vaables such as he coagulan concenaon n he bah s he coagulan concenaon n he polyme soluon n he empeaue of he soluon n he bah s and he adus of a polyme soluon flamen R. he ange of he vaables s chosen o be close o wha acually occus n pacce. he esuls ae deved usng ou mahemacal model fo he "sof" we spnnng pocess n he PAN-DMF-wae sysem. Dung a smulaon un one of he vaables s changed and he emanng ones ae held consan. Ineesngly he cuve of he oal gelaon me ploed agans fbe adus s a quadac paabola. hs fac confms he possbly of usng elaons of he ype 15. ONLUSION In hs pape we have come up wh a model fo gelaon n he we spnnng of synhec fbes fom polymes. In conas o s pevous counepas [1-7] makes possble he compuaon of gelaon knecs on he bass of a phase dagam he jon hea and mass ansfe fom he movable neface and mass ansfe coeffcens as funcons of empeaue. he model and he assocaed compue sofwae can also come n useful n calculang we spnnng vaables fo ohe polyme soluons. AKNOWLEDGMENS he pape was suppoed by he Russan Foundaon fo Basc Reseach pojec no A. A R 1 R ; a-hemal dffusvy; -concenaon; NOAION c -coagulan concenaon a he nsan of he phase anson a empeaue ; p -appoxmae soluon of he lnea poblem of dffuson; D-dffusvy; ef - eo funcon; ef -1 - nvese of he eo funcon; G - specfc hea; q m - hea of mxng pe un mass of coagulan; q v - hea of mxng pe un volume of coagulan; R o R fbe adus coodnae vayng wh me

14 fom R o R g ; R R - gelaon fon coodnae vayng wh me fom R o ; - empeaue a me a pon wh coodnae ; n - nal empeaue of polyme soluon; λ - coagulang bah empeaue; g - complee gelaon me; v - fbe velocy n coagulang bah; α - hea ansfe coeffcen; γ - volumec conacon coeffcen of polyme upon gelaon; λ - hemal conducvy; p - densy; θ c - s / n - s ; /R ; B - Bo numbe; Fo - Foue numbe. SUBSRIPS 1 fo coagulan; 2 3 fo solven; 2 fo R > >R; 13 fo < < R: j fo R >>R ; j 1 fo < <R; n - nal value of a quany; s - value of a quany a fbe suface n coagulang bah; k - k-h me neval n numecal soluon; c - value of a quany a a phase anson; g - quanes peanng o he polyme gel; l - quany deved fom solvng a lnea poblem; p - appoxmae soluon of a lnea dffuson poblem. REFERENES 1. Zyabsk A. eoecheske osnovy fomovanya volokna heoecal Foundaons of Fbe Spnnng Moscow: Khmya Sekov. A.. Kudyavsev. G.I.. and Kozhevnkov Yu.R eova fomovanya khmcheskkh volokon

15 heoy of Synhec Fbe Spnnng. Moscow: Khmya Kabosepnye snecheske volokna abon han Synhec Fbes Moscow: Khmya Gelle B.E. and Zakov E. Sudy of Fbe Spnnng Mechansms Khm. Volokna 1963 no. 3 p Heye H.J. and Gobe. V.. Fasefoschung und exlechnk M no. 12 p. 577; M no. 1 p. 33; M no. 7. p. 313; M no. 9 p Sekov A.. Kudjavcev. G.J. Klmenko V.S. Sekova L.A. and Kozevnkov. Ju.R Fasefoschung und exlechnk 1969 no. 3 p Sekov A.. and Budnsk G.A. Polyacylonle Fbe Spnnng by he Rhodande Pocess Khm. Volokna 1993 no. 5 p Kalabn A.L. Pakshve. E.A.. and Kukushkn N.A. Mahemacal Modelng of Flamen Fomaon n We Spnnng of Synhec Fbes Absacs of Papes 9 Mezhdunaodnaya konfeensya chas' 4: Maemacheske meody v khm khmchesko ekhnolog 9h In. onf. pa 4: Mahemacal Mehods n hemsy and hemcal Engneeng. ve: ve Gos. ekh. Unv p Shh. D. Numecal Heal ansfe. Washngon: Hemsphee Belyaev N.M. and Ryadno A.A. Meody eo eplopovodnos Mehods of Hea onducon heoy Moscow: Vysshaya Shkola vol Lykov A.V. eoya eplopovodnos heoy of Hea onducon Moscow: Vysshaya Shkola Yankov V.I. Pevadch.uk V.P. and Boyachenkov V.I. Posessy peeabok voloknoobazuyushchkh polmeov Pocessng of Fbe-fomng Polymes Moscow: Khmya speman V.L. Soluon Spnnng of Polyacylonle Fbes n Vaous Solvens. and. Sc. Dsseaon Kalnn Pakshve E.A. On Phase Dagams of enay Sysems Polyacylonle-Dmehyl Fomamde- Wae and Polyacylonle-Sodum Rhodanae-Wae Absacs of Papes // Vsesoyuznava konfeensya po emodnamke ogancheskkh soednen II All-Unon onf. on hemodynamcs of Oganc ompounds Go'k: Gok. Gos. Unv p Mezhov M.S.. A New echnque o Invesgae he Rae of Gelaon n We-Spun Fbes Khm. Volokna no. 3 p Venk A.I. Pblzhenny asche posessov eplopovodnos Appoxmae alculaon of Hea onducon Pocesses. Moscow: Gosenegozda