POLYMER MELT FLOW IN SUDDEN EXPANSIONS: THE EFFECTS OF VISCOUS HEATING

POLYMER MELT FLOW IN SUDDEN EXPANSIONS: THE EFFECTS OF VISCOUS HEATING P. S. B. Zdansk a, M. Vaz. Jr. a, and A. P. C. Das a a Unversdade do Estado de Santa Catarna Departamento de Engenhara Mecânca Barro Bom Retro CP. 89223, Jonvlle, Santa Catarna, Brasl zdansk@jonvlle.udesc.br ABSTRACT Sudden expanson s a typcal geometry found at the entrance of a mould cavty and s generally known n ndustry as gate. Flow smulaton of ths class of problems poses some dffcultes owng to couplng of momentum and energy laws, exstence of re-crculaton and steep pressure varatons. Ths work focuses on the physcal analyss of the lamnar and ncompressble polymer melt flow nsde channels wth sudden expansons amed at mappng the vscous heatng effect. The mathematcal model comprses the mass, momentum and energy conservaton laws. The pressure-velocty couplng s treated on solvng a Posson equaton for pressure. The Cross consttutve model s adopted to descrbe the non- Newtonan behavor of the flow. The governng equatons are dscretzed usng the fnte dfference method based on central, second order accurate formulae for both convectve and dffusve terms. Artfcal smoothng terms are added to control the odd-even decouplng problem. The man results demonstrate that the flow parameters, such as pressure drop along the channel and Nusselt number at the walls, are affected by vscous heatng. It has been found that, n such problems, any relable soluton must account for the non-sothermal effects. Keywords: non-newtonan flows, numercal analyss, sudden expansons, vscous heatng NOMENCLATURE c p specfc heat at constant pressure, J/(kg.K) h channel thckness at the entrance, m h c convectve coeffcent, W/(m 2.K) k thermal conductvty, W/(m.K) n power-law ndex, non-dmensonal Nu Nusselt number, Nu=h c h/k p statc pressure, Pa Re Reynolds number, Re=ρu n h/η T temperature, K t tme, s u, u j velocty components, m/s x, x j Cartesan coordnates, m Greek symbols & γ equvalent shear rate, s -1 η apparent vscosty, Pa.s η 0 Newtonan vscosty, Pa.s λ materal parameter, s ρ flud densty, kg/m 3 Subscrpts n w nlet secton wall INTRODUCTION Polymer melt flow s frequently found n many ndustral applcatons such as njecton moldng processes. The numercal smulaton of ths class of problems has ganed wdespread attenton n the last years. New mathematcal models, realstc rheologcal descrptons and numercal schemes are ongong research topcs pursued by the scentfc communty. In njecton moldng, a narrow channel s placed between the feedng channel and the mould cavty, as llustrated n Fg. 1. Such contracton s usually known n ndustry as gate and has, among others, the purpose of heatng the melt to best ft the cavty. The heatng effect s beleved (emprcally) to be attaned due to hgh vscous dsspaton of polymer melts. In ths context, the present work ams at analyzng flow features of polymer melts nsde channels wth sudden expansons, specally the non-sothermal effects. The lterature n ths subject s recent but stll scarce, beng specally devoted on studyng sothermal flows. New numercal methodologes to solve the problem were proposed n Bao (2002) and Mssrls et al. (1998). The former work adopts fnte elements whereas the latter uses the fnte volume method, both of whch wthn a framework of a generalzed Newtonan formulaton. Physcal analyss of the flow bfurcaton phenomenon was dscussed by Engenhara Térmca (Thermal Engneerng), Vol. 7 N o 01 June 2008 p. 65-70 65

Manca and De Bortol (2004), Neofytou and Drkaks (2003), Ternk et al. (2006) and Neofytou (2006). The references adopt the generalzed Newtonan formulaton wth an sothermal power-law model to descrbe the non-newtonan flow behavor. Pressure drop and vortex length n sudden expansons are flow features studed n Pnho et al. (2003). The authors dscuss the effects of the power-law ndex n sothermal flows. Ntn and Chhabra (2005) present a parametrc analyss of non-sothermal power-law flud flow past a rectangular obstacle. The drag coeffcent and Nusselt number are obtaned as functons of Reynolds number, power law-ndex and Peclet number. The soluton obtaned by the authors corresponds to Reynolds number rangng from 5 to 40, whch are typcal values for polymerc aqueous soluton, but too hgh for polymer melts. The present work dscusses physcal aspects of polymer melt flow n sudden expansons. The analyss assesses the nfluence of vscous heatng phenomenon on flow parameters such as pressure drop along the channel and Nusselt number at the walls. The numercal scheme s based on the technques orgnally conceved to solve Newtonan flows (Zdansk et al., 2004). The capablty of the scheme to handle non-newtonan flows nsde plane channels was demonstrated n Zdansk and Vaz Jr. (2006a, 2006b), beng straghtforward ts extenson to solve sudden expanson flows (Vaz Jr. and Zdansk, 2007). The results obtaned show that the effects of vscous heatng on flow parameters are relevant as the Reynolds number ncreases. Feedng Channel Gate Specmen Fgure 1. Typcal gate as t appears n a mould cavty THEORY The fully coupled Naver-Stokes and energy equatons, wthn a framework of generalzed Newtonan formulaton, s the mathematcal model adopted,.e., ( ρu ) ( ρu ju ) t + j p = + j ( ) u u j η T, & γ +, (1) j ( ρc T) ( ρc u T) p t + p = T k + η x ( T, & γ) & γ 2, (2) where η s the apparent vscosty, Pa/s -1, T the temperature, K, and γ & the equvalent shear rate, s -1. The remanng symbols n the precedng equatons are completely standardzed n lterature and gven by: p s the pressure, Pa; ρ the flud densty, kg/m 3 ; u and u j the velocty components, m/s; c p the specfc heat at constant pressure, J/kgK; x and x j the Cartesan coordnates, m; t the tme, s. The smulatons performed n ths work are for a commercal polymer Polyacetal POM-M90-44, whose non-newtonan behavor s accounted for the Cross consttutve model as η ( T, γ) & 0, (3) = 1.0+ η 1.0 n( T) [ λ( T) & γ] n whch η 0 s the Newtonan vscosty, Pa/s -1, n s the power-law ndex, non-dmensonal, and λ s a materal parameter, s, whose temperature dependence s expressed by the Arrhenus law. For more detals about the governng equatons and consttutve model the reader s referred to Zdansk and Vaz Jr. (2006a, 2006b). The ncompressble flow approach employed to derve the numercal approxmaton requres use of a pressure-velocty couplng scheme. The method used n ths work follows the same route presented n Zdansk et al. (2004), where a Posson equaton for pressure s solved to assure dvergence-free velocty feld. NUMERICAL METHOD As the emphass of the present work s the physcal analyses of mportant aspects of polymer melt flow n sudden expansons, only a bref dscusson on the numercal method s gven. Central fnte dfference formulae were used to dscretze both convecton and dffuson terms of the governng equatons. Therefore, the scheme s second order accurate n space. The varables are arranged n a colocated mesh, and artfcal vscosty terms are added externally to control the odd-even decouplng problem. The method s mplct and follows a pseudo-transent march n tme, amng at achevng the steady state soluton. The present methodology was orgnally proposed to solve Newtonan flows (Zdansk et al., 2004), and has been successfully appled to non-newtonan flows by Zdansk and Vaz Jr. (2006a, 2006b) and Vaz Jr. and Zdansk (2007). 66 Engenhara Térmca (Thermal Engneerng), Vol. 7 N o 01 June 2008 p. 65-70

RESULTS AND DISCUSSION The non-newtonan flow n sudden expansons, despte the smplcty of the geometry, exhbts some complex flud flow behavor. The presence of recrculaton regons and steep property varatons mpose dffcultes leadng to a demandng computatonal case. In the practcal sense, a flow expanson s typcally found at the gate and ts physcal understandng s mportant n ndustral njecton moldng processes. Fgure 1 represents a typcal gate located at the entrance of a mould cavty. The expanson secton can be clearly seen and the problem under dscusson s of great nterest n the practcal sense as well as n a scentfc ground. The geometry smulated n the present work s the 1:2 asymmetrc expanson, beng the channel length 3h and 20h upstream and downstream of the expanson secton respectvely (h s the channel thckness of the entrance secton and s assumed h = 4mm). The geometry, wth ts man dmensons, s represented at Fg. 2. The computatonal mesh adopted s non-unform wth ponts clusterng n the regons near the corner and sold walls. A total of 41 x 81 and 151 x 161 grd ponts are used for mappng the channel regons upstream and downstream of the expanson secton, respectvely. The boundary condtons used n the smulatons are the followng: no-slp condton for velocty components and prescrbed temperature at sold walls; unform velocty and temperature profles at the entrance and parabolc extrapolaton (non-reflexve) at the ext secton; for pressure, a lnear varaton s mposed at the entrance secton whereas the parabolc condton s enforced at both sold walls and ext secton. 3 h 20 h T w T w = T n = 453.0K and T n = 473.0K > T w = 453.0K. The followng thermal problems are addressed: smulaton usng () the complete energy equaton as t appears n Eq. (2), and () the energy equaton wthout vscous dsspaton (wthout the last term of Eq. (2)). The precedng strategy ams at assessng the nfluences of vscous heatng n such flows for the present Reynolds number range. Some general features of the flow topology are presented for T n (473.0K) > T w (453.0K) n Fgs. 3 6. The results correspond to the soluton of thermal problems () and () where nlet velocty s set constant u n =10cm/s. Streamlnes at the neghborhood of the step are represented at Fg. 3, n whch, a small vortex zone appears n the regon at the corner. The bubble s extremely confned at the concave corner due to the hgh flud vscosty of polymer melts. Ths flow topology s qualtatvely n accordance wth Bao s results for sothermal flows (Bao, 2003). The re-crculaton length s approxmately x/h = 0.25. It s nterestng to menton that the flow topology s very smlar for both thermal problems () and (), beng the dfferences n the vortex length vrtually neglgble. Ths feature s credted to the extremely low Reynolds number of the problem. Fgure 3. Streamlnes at the expanson regon for thermal problem () U n T n h y x 2 h y/h 1.60 1.20 0.80 T w Fgure 2. Sudden expanson geometry wth ts man dmensons The am of the present work s to nvestgate the nfluence of vscous heatng on expanson flows. The nlet velocty, u n, s vared from 4cm/s to 50cm/s. Ths velocty range renders Reynolds numbers around 10-4 10-3. The temperatures enforced at walls and nlet secton correspond to two set of values,.e, 0.40 x/h = 0.25 wth vscous heatng x/h = 0.25 wthout vscous heatng x/h = 1 wth vscous heatng x/h = 1 wthout vscous heatng x/h = 3 wth vscous heatng x/h = 3 wthout vscous heatng 0.04 0.08 0.12 0.16 u/un Fgure 4. Velocty profles from y/h = 0 (lower wall) to y/h = 2 (upper wall) Engenhara Térmca (Thermal Engneerng), Vol. 7 N o 01 June 2008 p. 65-70 67

Velocty, Temperature and vscosty profles at selected statons x/h (where x s measured from the expanson secton) are plotted n order to evaluate the dfferences between two thermal problems smulated. The attenton s focused on the neghborhood of the expanson secton, e.g., x/h = 0.25; 1.00; 3.00. The profles shown at Fgs. 4 6 hghlght that vscous heatng has lttle effect on the property dstrbutons for nlet veloctes lower than 10cm/s (u n 10cm/s and h = 4mm). It has been found that vscous heatng promotes a maxmum temperature rse of 0.9K for the condtons smulated (around y/h = 0.25 for x/h = 1.0). Further dscussons regardng the shape of the profles are addressed n authors prevous works for plane channels (Zdansk and Vaz Jr. 2006a, 2006b) and expanson flows (Vaz Jr. and Zdansk, 2007). y/h 1.60 1.20 0.80 0.40 x/h = 0.25 wth vscous heatng x/h = 0.25 wthout vscous heatng x/h = 1.00 wth vscous heatng x/h = 1.00 wthout vscous heatng x/h = 3.00 wth vscous heatng x/h = 3.00 wthout vscous heatng The mean pressure drop from nlet to ext channel sectons for the two thermal problems smulated and dfferent nlet veloctes s shown n Fg. 7. The temperatures adopted as boundary condton at entrance secton and wall are T n = T w = 453.0K and T n = 473.0K > T w = 453.0K. The results are presented as functon of Reynolds number, ρu n h Re=, (4) η where the mean vscosty (η ) s estmated at the channel ext secton. Clearly, the fgure shows that vscous heatng effect s gettng more pronounced as veloctes are ncreased (hgher Reynolds numbers). For lower veloctes (u n < 20cm/s), the maxmum dfference of pressure losses s 1% for thermal problems () and (). Besdes, one assesses that pressure drop s ever lower for thermal problem (), especally at hgher Reynolds number. Ths s a consstent result snce at hgher Reynolds number vscous heatng promotes an overheatng causng the vscosty to decrease renderng lower pressure losses. When comparng the solutons for the two sets of nlet temperatures adopted one assesses a reducton of around 34% n pressure losses for hgher Reynolds number when nlet temperature s rased by 20K. Ths result demonstrates the great nfluence of temperature on pressure varatons and consequently on power requred to pump the polymer nsde a mould cavty. 5.00E+6 45 455.00 46 465.00 47 475.00 T [K] Fgure 5. Temperature profles from y/h = 0 (lower wall) to y/h = 2 (upper wall) 1.60 1.20 x/h = 0.25 wth vscous heatng x/h = 0.25 wthout vscous heatng x/h = 1 wth vscous heatng x/h = 1 wthout vscous heatng x/h = 3 wth vscous heatng x/h = 3 wthout vscous heatng Pressure drop [Pa] 4.00E+6 3.00E+6 E+6 1.00E+6 Wth vscous heatng (Tn = Tw) Wthout vscous heatng (Tn = Tw) Wth vscous heatng (Tn > Tw) Wthout vscous heatng (Tn > Tw) y/h 0.80 0.40 E+0 E+0 1.00E-3 E-3 3.00E-3 4.00E-3 Reynolds Fgure 7. Mean pressure drop from nlet to ext channel sectons 40 80 120 160 Vscosty [Pa.s] Fgure 6. Vscosty profles from y/h = 0 (lower wall) to y/h = 2 (upper wall) The local Nusselt number dstrbutons for thermal problems () and () are presented at Fg. 8. The followng Nusselt defnton s adopted, h ch Nu=, (5) k 68 Engenhara Térmca (Thermal Engneerng), Vol. 7 N o 01 June 2008 p. 65-70

where the convectve coeffcent s defned as h c q" s =. (6) (T T ) w m In the precedng expresson T m s the classcal bulk temperature and q " s the heat flux evaluated at the lower wall (y/h = 0). The results correspond to nlet temperature T n = 473.0K > T w = 453.0K for two dstnct nlet veloctes u n = 6cm/s and u n = 18cm/s. Once agan the results clearly evnce that vscous heatng becomes more mportant for ncreasng veloctes. Notceably, for nlet velocty 6cm/s, the Nusselt number for problem () s around 4% hgher than problem () (for the range x/h > 2.0). Otherwse, when nlet velocty s rased to u n = 18cm/s, the dfferences between the two thermal problems become around 15% (for x/h > 2.0). Besdes, the Nusselt number s always hgher when vscous heatng s consdered, an expected result. It s nterestng to note that the general trend of the Nusselt curve for the present polymer s smlar to the Newtonan flow n sudden expansons (Zdansk at al., 2005),.e., there s a local maxmum near the reattachment pont, followed by a slow decrease as the flow recovers the plane channel tendency. Nusselt 7.00 6.00 5.00 4.00 3.00 1.00 6 cm/s wth vscous heatng 6 cm/s wthout vscous heatng 18 cm/s wth vscous heatng 18 cm/s wthout vscous heatng 4.00 8.00 1 16.00 2 x/h Fgure 8. Nusselt number dstrbuton along the channel length CONCLUSIONS The Polymer melt flow n sudden expansons s studed numercally. The generalzed Newtonan formulaton s adopted, beng the non-newtonan behavor of the flow descrbed by the Cross consttutve relaton. The numercal method was able to handle successfully the strong non-lnearty and coupled character of the problem. The results obtaned clearly evnce the effects of vscous heatng on pressure losses and Nusselt number for ths class of flow. The fndngs may be summarzed as follows: (a) t has been found that vscous heatng promotes a maxmum temperature rse of 0.9K for nlet veloctes smaller than 10cm/s; t s worth to menton that the range of veloctes smulated n ths work s typcally found n ndustral njecton problems; therefore, the fndngs may be relevant for technologcal feld; (b) mean pressure drop n the channel s strongly affected by the nlet temperature; otherwse, the vscous heatng exhbt only moderate nfluence; for the range smulated (u n < 50cm/s), the maxmum dfference of pressure losses s found to be 4% for thermal problems () and (); n addton, an nlet temperature rse of 20K leads to a reducton n the mean pressure drop of around 34% for both thermal problems; (c) the Nusselt number s more senstve to the vscous heatng effect than to pressure losses; when nlet velocty s ncreased to 18cm/s the dfference between two thermal problems becomes 15%. ACKNOWLEDGEMENTS The co-author A. P. C. Das gratefully acknowledges the PIBIC/CNPq/UDESC fnancal support program. REFERENCES Bao, W., 2003, An economcal fnte element aproxmaton of generalzed Newtonan flows, Computer Methods n Appled Mechancs and Engneerng, Vol. 191, pp. 3637-3648. Koh, Y. H., Ong, N. S., Chen, X. Y., Lam, Y. C. and Cha, J.C., 2004, Effect of temperature and nlet velocty on the flow of a non-newtonan flud, Internatonal Communcatons n Heat and Mass Transfer, Vol. 31, No. 7, pp. 1005-1013. Neofytou, P., 2006, Transton to asymmetry of generalzed Newtonan flud flows through a symmetrc sudden expanson, Journal of Non- Newtonan Flud Mechancs, Vol. 133, pp. 132-140. Neofytou, P. and Drkaks, D., 2003, Non- Newtonan flow nstablty n a channel wth a sudden expanson, Journal of Non-Newtonan Flud Mechancs, Vol. 111, pp. 127-150. Ntn, S. and Chhabra, R. P., 2005, Nonsothermal flow of a power law flud past a rectangular obstacle (of aspect rato 1 x 2) n a channel: Drag and heat transfer, Internatonal Journal of Engneerng Scence, Vol. 43, pp. 707-720. Manca, R., and De bortol, A. L., 2004, Smulaton of sudden expanson flows for power law fluds, Journal of Non-Newtonan Flud Mechancs, Vol. 121, pp. 35-40. Mssrls, K. A., Assmacopoulos, D. and Mtsouls E., 1998, A fnte volume approach n the smulaton of vscoelastc expanson flows, Journal of Non-Newtonan Flud Mechancs, Vol. 78, pp. 91-118. Engenhara Térmca (Thermal Engneerng), Vol. 7 N o 01 June 2008 p. 65-70 69

Pnho, F. T., Olvera, P. J. and Mranda, J. P., 2003, Pressure losses n the lamnar flow of shearthnnng power-law fluds across a sudden axsymmetrc expanson, Internatonal Journal of Heat and Flud Flow, Vol. 24, pp. 747-761. Ternk, P., Marn, J.and Zunk, Z., 2006, Non- Newtonan flud flow through a planar symmetrc expanson: Shear-thckenng fluds, Journal of Non- Newtonan Flud Mechancs, Vol. 135, pp. 136-148. Vaz Jr., M. and Zdansk, P. S. B., 2007, A fully mplct fnte dfference scheme for velocty and temperature coupled solutons of polymer melt flow, Communcatons n Numercal Methods n Engneerng, Vol. 23, pp. 285-294. Zdansk, P. S. B. and Vaz Jr., M., 2006a, Polymer melt flow n plane channels: effects of the vscous dsspaton and axal heat conducton, Numercal Heat Transfer Part A: Applcatons, Vol. 49, No. 2, pp. 159-174. Zdansk, P. S. B. and Vaz Jr., M., 2006b, Polymer melt flow n plane channels: hdrodynamc and thermal boundary layers, Journal of Materals Processng Technology, Vol. 179, pp. 207-211. Zdansk, P. S. B., Ortega, M. A. and Fco Jr., N. G. C. R., 2004, Numercal smulaton of the ncompressble Naver-Stokes equatons, Numercal Heat Transfer Part B: Fundamentals, Vol. 46, No. 6, pp. 549-579. Zdansk, P. S. B., Ortega, M. A., Fco Jr., N. G. C. R., 2005, Heat transfer studes n the flow over shallow cavtes, Journal of Heat Transfer, Vol. 127, pp. 699-712. 70 Engenhara Térmca (Thermal Engneerng), Vol. 7 N o 01 June 2008 p. 65-70