Estimation of elongational viscosity of polymers from entrance loss data using individual parameter...

Estimation of ElongationalViscosity of Polymers fromEntrance Loss Data UsingIndividual ParameterOptimization

MAHESH GUPTAMechanical Engineering–Engineering Mechanics Department, Michigan Technological University,Houghton, Michigan 49931

Received: February 22, 2001Accepted: January 28, 2002

ABSTRACT: The elongational viscosity model proposed by Sarkar and Gupta(Journal of Reinforced Plastics and Composites 2001, 20, 1473), along with theCarreau model for shear viscosity is used for a finite element simulation of theflow in a capillary rheometer. The entrance pressure loss predicted by the finiteelement flow simulation is matched with the corresponding experimental data topredict the parameters in the elongational viscosity model. To improve thecomputational efficiency, various elongational viscosity parameters are optimizedindividually. Estimated elongational viscosity for a Low Density Polyethylene(Dow 132i) is reported for two different temperatures. C© 2002 Wiley Periodicals,Inc. Adv Polym Techn 21: 98–107, 2002; Published online in Wiley Interscience(www.interscience.wiley.com). DOI 10.1002/adv.10017

Introduction

T he generalized Newtonian constitutive mod-els1 with shear-thinning viscosity have been

Correspondence to: Mahesh Gupta; e-mail: [email protected].

commonly employed to simulate polymeric flows.Even though the generalized Newtonian modelscan accurately simulate the shear-dominated flowssuch as those encountered in plastic molding pro-cesses, the simulation of an elongation-dominatedflow, such as the flow in extrusion dies, with thissimplified approach can have a large error. Theprimary reason for the poor prediction of an

Advances in Polymer Technology, Vol. 21, No. 2, 98–107 (2002)C© 2002 Wiley Periodicals, Inc.

ESTIMATION OF ELONGATIONAL VISCOSITY OF POLYMERS FROM ENTRANCE LOSS DATA

elongation-dominated flow is the low elongationalviscosity of the generalized Newtonian models.1 Thelong chain polymer molecules exhibit stiff resistanceto an elongational deformation. Therefore, the elon-gational viscosity of a polymer, which is definedas the ratio of the elongational stress to elonga-tional strain rate, is very high. An accurate knowl-edge of the elongational viscosity of a polymer isimportant for simulating the elongation-dominatedflows.

Various techniques such as uniaxial extension,lubricated compression, fiber spinning, bubble col-lapse, stagnation flow, etc., have been used inthe past to experimentally characterize the elon-gational viscosity of polymers. A good reviewof these techniques and the difficulties associatedwith each of these techniques is presented byMacosko.2 Among these, the most commonly usedtechnique is the uniaxial extension. An elongationalrheometer using uniaxial extension, which is basedupon Meissner’s work,3 is currently marketed byRheometric Scientific, Inc., Piscataway, New Jersey,(http://www.rheosci.com). However, maintaining asteady uniaxial extensional deformation is difficultat a high elongation rate.2 Therefore, Meissner’stechnique can be used to determine elongational vis-cosity only at very low elongation rates (<10 s−1).Also, Meissner’s technique can only be used forhighly viscous melts. Since the deformation ratein polymer processing can range from 1000 to100,000 s−1, this technique cannot provide the elon-gational viscosity data required for simulation ofmost polymer processing techniques.

Because of the difficulties associated with directmeasurement of elongational viscosity of a polymer,the flow in a channel with abrupt contraction (en-trance flow) has often been used for an indirect mea-surement of elongational viscosity.2 A fluid goingthrough an abrupt contraction experiences a rapidelongational deformation. Therefore, if the fluid hasa high elongational viscosity, to maintain the elon-gational flow, a large pressure gradient is requirednear the contraction. Accordingly, for a polymer go-ing through an abrupt contraction, a steep pressuredrop, called entrance pressure loss is encounterednear the abrupt contraction. The value of entrancepressure loss, which depends upon the flow ratein the channel, can be used for an indirect mea-surement of the strain-rate dependence of the elon-gational viscosity of a polymer. The advantage ofthe entrance flow method for measuring elonga-tional viscosity is its applicability at high elongationrates. Furthermore, an existing capillary rheometer,

which is commonly used for measuring shear vis-cosity, can also be used for elongational viscosityestimation.

Assuming that the pressure loss in an entranceflow can be obtained by summing the pressuredrops due to shear and elongational deformations,Cogswell4 obtained separate expressions for thesetwo pressure drops. To further simplify the analy-sis, Cogswell made several other assumptions listedbelow:

1. Shear stress (τ ) and shear rate (γ̇) are related bypower-law model (τ = Aγ̇n);

2. Elongational viscosity is constant;3. The interface between the recirculation region

and the main flow is a conical surface (funnel-shaped flow);

4. No-slip on the interface between recirculationregion and the main flow;

5. Fully developed flow with zero radial velocity;6. Neglect Weissenberg–Robinowitsch correcti-

on1;7. Isothermal, inertia-less, incompressible flow

with no normal stresses in shear.

With these assumptions, Cogswell performed a forcebalance on a differential section of the funnel-shapedentry region near abrupt contraction. By integratingthe expressions thus obtained, Cogswell obtainedthe following expressions for average elongationrate (ε̇a) in entrance flow and the correspond-ing elongational viscosity (ηe) for an axisymmetricflow.

ε̇a = 4τwγ̇a

3(n+ 1)1pe(1)

ηe = 3(n+ 1)1pe

8ε̇a(2)

where τw = 1pR/2L is the shear stress at thecapillary wall, γ̇a = 4Q/(πR3), 1p, and Q are respec-tively, the pressure drop and flow rate in the capil-lary, R and L are the radius and length of the capi-llary, respectively, and 1pe is the entrance pressureloss.

By removing some of the assumptions in theCogswell’s analysis, Binding5 obtained a more ac-curate expression for entrance pressure loss in anabrupt contraction. However, many of the assump-tions in Cogswell’s analysis, listed below, are re-tained in Binding’s analysis:

ADVANCES IN POLYMER TECHNOLOGY 99


1. Power-law model for shear viscosity (ηs =Aγ̇n−1) as well as for elongational viscosity(ηe = Bε̇m−1);

2. The interface between the recirculation regionand the main flow is a conical surface (funnel-shaped flow);

3. No-slip on the interface between recirculationregion and the main flow;

4. Fully developed flow with no radial velocity;5. |dR/dz| ¿ 1, such that the terms involving

(dR/dz)2 and d2 R/dz2 can be neglected, imply-ing a large recirculation zone;

6. Energy consumed in recirculation zone can beneglected;

7. Isothermal, inertia-loss, incompressible flowwith no normal stresses in shear.

With these simplifications, Binding employed en-ergy principles to obtain the following equation forentrance pressure loss:

1pe = 2A(1+m)2

3m2(1+ n)2

[mB (3n+ 1)nm Inm

A

] 1m+1

× γ̇wm(n+1)

m+1[1− α 3m(n+1)

m+1]

(3)

where α = R0/R1, with R0 and R1 being theradii of downstream and upstream channels re-spectively, Inm =

∫ 10 |((3n+ 1)ρ1+1/n/n)− 2|m+1ρdρ,

and the shear rate at capillary wall γ̇w = (3n+1)Q/(πnR3

0). If the power-law model for shear vis-cosity of a polymer, and entrance pressure loss isknown for two different flow rates in an abrupt con-traction, Eq. (3) can be used to determine the power-law model for the elongational viscosity (B, m) of thepolymer.

By using independent power-law models forshear and elongational viscosities, Gupta6 analyzedthe effect of elongational viscosity on vortex forma-tion and entrance pressure loss in an axisymmet-ric 4:1 entrance flow. Similar analyses of the flowin a channel with abrupt contraction for the planar7

and the three-dimensional8 cases have also been pre-sented. The truncated power-law model was usedfor the shear viscosity as well as for the elongationalviscosity in the axisymmetric and planar cases.6−8

This approach for simulating polymeric flows isframe invariant. Also, if irrespective of the valueof eII, the specified shear and elongational viscos-ity models are such that ηe(eII) = βηs(eII), with β = 3for axisymmetric flows, β = 4 for planar flow, and eII

being the second invariant of the strain-rate tensor,then for any three-dimensional flow, the velocity andpressure distributions predicted by this approach areidentical to those from the generalized Newtonianformulation. It should be noted that ηe(eII) = βηs(eII)is satisfied by all generalized Newtonian models.More recently, using the same approach, Sarkarand Gupta9 employed the Carreau model for shearviscosity

ηs = η0(1+ (λeII)2)n−1

2 (4)

and the following modification of the Carreau modelfor elongational viscosity

ηe = η0

[3+ δ

{1− 1√

1+ (λ1eII)2

}][1+ (λ2eII)2]

m−12

(5)

For δ = 0, the elongational viscosity model in Eq. (5),reduces to the Carreau model with the elongationalviscosity parameters λ2 and m, respectively, replac-ing λ and n in the shear viscosity model. It is ev-ident from Fig. 1 that the parameter λ1 in Eq. (5)specifies 1/eII for transition between Newtonianand elongation-thickening portions of the viscos-ity strain–rate curve, whereas δ characterizes the to-tal increase in viscosity in the elongation-thickeningportion. Parameters λ2 and m in Eq. (5) specify1/eII for transition between elongation-thickeningand power-law regions, and the power-law indexfor elongational viscosity, respectively.

In this work, we have used the Carreau model forshear viscosity [Eq. (4)] and the model proposed bySarkar and Gupta for elongational viscosity [Eq. (5)].Knowing the shear and elongational viscosity pa-rameters, a finite element simulation of the entranceflow is performed. The finite element simulation ofthe entrance flow eliminates most of the simplify-ing assumptions of Cogswell’s and Binding’s anal-yses. However, the flow simulation still assumes anisothermal, inertia-less, incompressible flow with nonormal stresses in a shear flow. With a knowledgeof the shear viscosity and entrance pressure loss,using the procedure presented in the next section,the elongational viscosity parameter are optimizedsuch that the difference between the experimentalvalues of entrance pressure loss and the correspond-ing predictions from the finite element simulation isminimized.

100 VOL. 21, NO. 2


(a) (b)

(c) (d)

FIGURE 1. Effect of various parameters in Eq. (5), λ1 (a), δ (b), λ2 (c), and m (d), on elongational viscosity.

Estimation Procedurefor Elongational ViscosityParameters

Earlier we reported a minimization procedure forsimultaneous optimization of all four elongationalviscosity parameters (δ, λ1, λ2, and m).10 In particu-lar, starting with an initial guess for the four param-eters, the simplex optimization scheme was used toiteratively improve their values such that the dif-ference between experimental and predicted valuesof entrance loss is minimized. Even though such aglobal minimization procedure is more rigorous, andattempts to minimize the error over the complete

range of flow rate vs entrance loss data, it requiresvery large amount of computational time, and con-vergence to the optimal values is prohibitively slowat times. It should be noted that the main reason forthe slow convergence is not the simplex optimiza-tion scheme, but the simultaneous optimization ofthe all four elongational viscosity parameters, whichrequires repeated simulation of the entrance flow atseveral flow rates and at various test values of thefour parameters.

To improve the efficiency of the estimation soft-ware, in this work, only the values of λ2 and mare optimized simultaneously, which is followed byindividual optimization of δ and λ1. It should benoted that parameters λ2 and m affect the elonga-tional viscosity only in the power-law region at high



strain rate, whereas δ and λ1 control the shape ofthe elongational-thickening portion of the viscos-ity curve (Fig. 1). In particular, λ1 determines thestrain rate for onset of elongation-thickening region,whereas δ controls the total increase in the viscosityin the thickening region. Since δ andλ1, andλ2 and maffect the elongational viscosity in different regions,values of δ and λ1, and those of λ2 and m can be opti-mized independently. Because of this localized effectof δ and λ1, and that of λ2 and m, if all the entrancepressure loss data is in the power-law region, opti-mization of the elongation-thickening parameters isdifficult and vice versa.

To optimize the four elongational viscosity pa-rameters, the experimental data for entrance loss isrequired for at least four different flow rates. Two ofthese points should be in the power-law region (todetermine λ2 and m). If the polymer melt exhibitsthe elongation-thickening region, one of the pointsshould be in the elongation-thickening region (todetermine δ) and the fourth point should bein the transition region between Newtonian andelongation-thickening behavior (to determine λ1).Typically, entrance pressure loss vs flow rate datais available at a large number of points (À4). Sincean entrance flow simulation for a fixed set of elon-gational viscosity parameters and flow rate requiressignificant amount of computation time (1–10 minon a Sun workstation, depending upon the num-ber of nodes in the finite element mesh and numberof iterations required for convergence), and the en-trance flow is repeatedly simulated at the entranceloss vs flow rate data points, if all the data pointsare used in the optimization scheme, a very largecomputation time is required for estimation of theelongational viscosity parameters. Therefore, a sys-tematic scheme, described in the next paragraph, isemployed in this work to select the four entranceloss vs flow rate data points for estimation of thefour elongational viscosity parameters.

Estimation procedure for elongational viscosityparameters starts with an initial guess for the fourparameters. The software allows the user to specifyan initial estimate of the four parameters. Otherwise,the software automatically initializes the elonga-tional viscosity parameters based upon the general-ized Newtonian formulation, that is ηe(eII) = 3ηs(eII),or the values obtained by the Binding’s analysis.5 Itshould be noted that for both of these cases δ = 0,andλ1 has no affect on elongational viscosity if δ = 0.Therefore, only λ2 and m are initialized. Since the ex-perimental data at higher flow rate is expected to bein the power-law region, the data point at the highest

flow rate and another data point with flow rate ofabout 1/100 of the highest flow rate are used for es-timation of power-law parameters λ2 and m. Oncethe two points for estimation of power-law param-eters are identified, as shown in Fig. 2, the optimalvalues of λ2 and m are determined by alternating theNewton–Raphson method for λ2 and m. The func-tion fm and fλ2 are the differences in experimentaland predicted values of entrance pressure loss for

FIGURE 2. Flow chart for optimization of elongationalviscosity parameters.

102 VOL. 21, NO. 2


points with the highest flow rate and 1/100 of thehighest flow rate, respectively. At this point of thealgorithm λ2 and m have been identified. However,δ = 0 and λ1 is also unknown. With δ = 0 and usingthe estimated value of λ2 and m for the elongationalviscosity curve, the data point having the largest dif-ference between experimental and predicted valueof entrance loss is used for optimization of δ. If someof the points in the entrance loss vs flow rate dataare in the Newtonian range of elongational viscosity,experimental value of entrance loss for these pointsshould match with the predicted entrance loss withδ = 0 in the elongational viscosity model [Eq. (5)].Therefore, starting with the lowest flow rate, thefirst data point with experimentally determined en-trance loss at least 10% greater than the correspond-ing prediction is used to optimize the value of λ1.Before starting the optimization of δ, the parameterλ1 is initialized to a very large value λ2 = λ1 × 1010,which implies that elongation thickening is assumedto start at a very low elongation rate. Using the pointidentified above for optimization of δ, as shown inFig. 2, Newton–Raphson method is used again to es-timate the value δ. The function fδ in Fig. 2 is thedifference between experimental and predicted en-trance loss for the point being used for optimizationof δ. Finally, the parameter λ1 is estimated by mini-mizing the entrance loss error for the point identifiedabove for λ1 optimization. The bracketing and bisec-tion method11 is used for optimizing the value of λ1.The Newton–Raphson technique cannot be used foroptimizing λ1, because a change in λ1 has no effecton entrance flow if the current value of λ1 is far awayfrom its optimal value. This insensitivity to λ1 is ev-ident from Fig. 1a, which shows that λ1 only affectsthe elongational viscosity curve near the onset of theelongation-thickening region.

It should be noted that during the optimizationof the parameter δ, if the value of δ is changed with-out altering the other elongational viscosity param-eters, the elongational viscosity curve in the power-law region is shifted horizontally along the eII axis(Fig. 1b). However, the effect of δ on the power-lawregion can be countered by changing the value of λ2.In this work, the value of λ2 is changed such that thepower-law region of the elongational viscosity curvebeyond the strain rate for which ηe = η0 remains un-affected as the value of δ is changed. It can be easilyshown that the strain rate for which ηe = η0 in thepower-law region is given by:

e∗II =1λ2p

√(3+ δp)2/(1−m) − 1 (6)

FIGURE 3. Effect of the coordinated change in δ and λ2using Eqs. (6) and (7) such that the elongational viscosityin the elongation-thickening region is modified, but theelongational viscosity in the power-law region remainsunchanged.

where λ2p and δp are the values of λ2 and δ in theprevious iteration. As the value of δ is changed to δn

in the current iteration, the power-law region of theelongational viscosity curve can be restored to thecurve in the previous iteration by changing λ2 to λ2n

given by the following equation:

λ2n = 1e∗II

√(3+ δn)2/(1−m) − 1 (7)

Figure 3 shows the effect of such a coordinatedchange in λ2 and δ given by Eqs. (6) and (7) on theelongational viscosity curve. It is evident from Fig. 3that only the elongational viscosity curve in the shearthickening region is affected if λ2 and δ are changedaccording to Eqs. (6) and (7).

Experimental Data

A Goettfert Rheometer 1000 was used to mea-sure the viscosity and entrance pressure loss. Thediameter of capillary in all the dies used was 1 mm.The pressure loss vs capillary length curves for dieswith capillary length of 5, 20, 30, and 40 mm wereextrapolated to obtain the entrance pressure loss.The Weissenberg–Robinowitch correction was ap-plied to account for the nonparabolic velocity profile.A Bohlin VOR with 25-mm parallel plates was usedto measure the shear viscosity at low shear rates. Inthe experiments, a Low Density Polyethylene (Dow



132i) was used. The entrance pressure loss and shearviscosity was measured for 160 and 175◦C.

Results and Discussion

For the results reported in our elongational vis-cosity estimation papers so far,10, 12 we used theP2 P1 (quadratic velocity, linear pressure) triangu-lar finite element. Even though the P2 P1 elementaccurately predicts the velocity and pressure distri-butions in an incompressible flow, it requires a rel-atively large amount of computation time to sim-ulate a flow. To estimate the elongational viscosity,since the entrance flow is simulated repeatedly atvarious flow rates and for various test values of theelongational viscosity parameters, computational ef-ficiency of the flow simulation is important in orderto be able to estimate the elongational viscosity in areasonable amount of time. To improve the efficiencyof the incompressible flow simulation, the P+1 P1 minielement of Arnold, Brezzi, and Fortin,13 with the sim-plified bubble function suggested by Coupez andMarie,14 has been employed for the results reportedin this paper. For a triangular finite element, thesimplified bubble consists of three piece-wise lin-ear functions.14 An advantage of the piece-wise lin-ear bubble is that it allows an exact integration overthe triangular finite element. Employing the staticcondensation technique,15 the velocity equationscorresponding to the bubble node were eliminatedat the element level, resulting in the higher com-putational efficiency of the mini element. To avoidthe bubble reconstruction, the nonlinearity in therheological behavior is treated by projection of theP+1 velocity field on the P1 subspace. To simulatea three-dimensional flow, Coupez and Marie,14 usedthe preconditioned conjugate residual (PCR) methodfor solving the set of simultaneous linear equations.The PCR method was also implemented in this work.However, for a small number of nodes, typicallyused in a two-dimensional finite element simula-tion, the direct solution of the linear equations byLU decomposition technique was found to be moreefficient than the PCR method. For small number ofnodes, better efficiency of the direct solver was alsoreported by Coupez and Marie.14 Therefore, a directsolver, using the LU decomposition technique wasused for obtaining the results reported here. The fi-nite element mesh used (Fig. 4) has 1633 nodes and3040 elements.

FIGURE 4. Finite element mesh used for the 12:1entrance flow simulation (a) over the complete flowdomain (b) in vicinity of entrant corner.

In this paper, the elongational viscosity for a Dow132i is estimated at two different temperatures.For Dow 132i at 160 and 175◦C, the experimentaldata for entrance pressure loss at various flow ratesis shown in Fig. 5 (F. A. Morrison, personal com-munication, 2000). As expected, the entrance lossincreases significantly as the flow rate is increased.At a fixed flow rate the entrance loss is higher forthe lower temperature. The shear viscosity for Dow132i, which is obtained by fitting the Carreau modelto the experimental data, is shown in Fig. 6. The

FIGURE 5. Entrance loss vs. flow rate for Dow 132i.

104 VOL. 21, NO. 2


FIGURE 6. Variation of shear (ηs) and elongational (ηe)viscosities of Dow 132i with the second invariant ofstrain-rate tensor (e II).

Carreau model parameters for the shear viscosity ofDow 132i are as follows (F. A. Morrison, personalcommunication, 2000):

η0 (Pa s) λ (s−1) n

160◦C 2.6 × 105 37.35 0.3409

175◦C 2.1 × 105 31.69 0.3409

These Carreau model parameters along withthe entrance loss data (shown in Fig. 5) were usedin this work to predict the elongational viscosityemploying the algorithm discussed earlier in thispaper. For the two different temperatures, thepredicted elongational viscosity is shown in Fig. 6.It should be noted that the predicted elongationalviscosity is plotted against eII =

√3ε̇ (Fig. 6). The

power-law parameters for the elongational viscositywere predicted using the entrance pressure lossvalues for flow rates of 5.65 and 339 mm3/s for160◦C and for 5.66 and 452 mm3/s for 175◦C. Withthese control points the predicted elongationalviscosity parameters are as follows:

λ2 (s−1) m

160◦C 1.85 0.28175◦C 2.85 0.32

It is noted that even though in good agreement,these values of λ2 and m are slightly different fromthe values reported in Ref. 12. This minor differencein the values of λ2 and m is expected because the

P2 P1 element was used for the results reported inRef. 12, whereas the P+1 P1 element has been em-ployed in this paper. Using these values of the twoelongation-thinning parameters, with δ = 0, that is,no elongation thickening, the largest difference inthe experimental and predicted entrance pressureloss was obtained at the lowest flow rates in Fig. 5,which are 1.13 and 0.565 mm3/s for 160 and 175◦C,respectively. In our earlier work,12 the attempt tooptimize δ using the entrance loss data at thesetwo points was unsuccessful, because the predictedentrance loss at these two points was found tobe insensitive to a change in δ. Since then, thesensitivity of entrance loss to δ was improved in thesoftware, which allowed us to optimize δ and λ1 inthis work. As δ is modified during the optimizationprocess, in order to keep the power-law region ofthe elongational viscosity curve unaltered, the valueof λ2 was adjusted according to Eqs. (6) and (7).The optimized values of the elongational viscosityparameters for the curves shown in Fig. 6 are asfollows:

δ λ1 (s−1) λ2 (s−1) m

160◦C 4.29 6549.7 6.5 0.28175◦C 7.58 1869.1 18.7 0.32

It is evident from Fig. 5, that with δ = 0, i.e. no elon-gation thickening, the predicted entrance losses atonly a few points near the lowest flow rates (onepoint for 160◦C and three points for 175◦C) are signif-icantly different that the corresponding predictionswith nonzero δ values. Since only a limited exper-imental data on entrance flow is available in theelongation-thickening region, the elongation thick-ening regions for 160 and 175◦C shown in Fig. 6may not be very accurate. This low accuracy of thepredicted elongation-thickening region is probablycausing the two elongational viscosity curves to in-tersect unexpectedly in the elongation-thickeningregion. The entrance pressure loss predicted by ageneralized Newtonian formulation using Carreaumodel, which is also shown in Fig. 5, is only about20% of the corresponding experimental data.

For Dow 132i, the elongational viscosities at 160and 175◦C, predicted by Cogswell’s and Binding’sanalyses are also shown in Fig. 6. Interestingly, eventhough Cogswell’s and Binding’s predictions arebased upon numerous simplifying assumptions, theelongational viscosity predicted by the two analysesare in good agreement with the predictions from thepresent work.



For three different flow rates in the range of theexperimental data in Fig. 5, unit vectors along thedirection of velocity near the abrupt contraction areshown in Fig. 7. In contrast to the results reported inRef. 6, the recirculation region near the abrupt con-traction shrinks as the flow rate is increased. For thevelocity distributions at various flow rates reportedin Ref. 6, the elongational power-law index (m) waslarger than the power-law index (n) for the shear vis-cosity. With m > n, the Trouton ration (Tr = ηe/ηs)increases with flow rate. Accordingly, the recirculat-ing vortex also grows with flow rate. In this work,for Dow 132i, since m is less than n, Trouton ra-tio decreases as the flow rate is increased. Because

(a) (b)

(c)

FIGURE 7. Recirculation in an axisymmetric 12:1 entrance flow of Dow 132i at 160◦C. The flow rates are 1 mm3/s (a),10 mm3/s (b), and 100 mm3/s (c).

of this decrease in Trouton ratio, for Dow 132i, thepredicted recirculating vortex shrinks as the flowrate is increased. The velocity distributions shown inFig. 7 are for 160◦C. Similar reduction in recirculationwith increasing flow rate was found for 175◦C. Also,the recirculation region and the pressure distribu-tion along the channel, predicted by the two differenttypes of finite elements, P2 P1 and P+1 P1, were foundto be in good agreement.

In closing, it is emphasized that the results re-ported here should be treated as estimates of theresistance of Dow 132i to an elongational deforma-tion. Even though these could be used as the esti-mated elongational viscosity of the polymer, because

106 VOL. 21, NO. 2


of the several simplifications in the present analy-sis, the intrinsic elongational viscosity of the poly-mer obtained by a direct elongation experiment, canbe somewhat different than these estimates. For in-stance, the first and second normal stress differences,which can also affect the entrance pressure loss, havenot been accounted in the present analysis. Eventhough viscous dissipations can change the temper-ature of the polymer melt in a capillary rheome-ter, the simulations reported here are isothermal.The entrance flow is sometimes susceptible to non-axisymmetric instabilities, which can only be cap-tured in a three-dimensional flow simulation. Also,the present analysis employs a steady state simula-tion. Therefore, transient flow instabilities, which aresometimes encountered in entrance flow, could notbe simulated in the present work. In spite of thesesimplifications, because of the numerous difficultiesassociated with the direct measurement of elonga-tional viscosity, the method employed here providesan attractive alternative for estimation of the elonga-tional viscosity of a polymer.

Conclusions

A scheme employing individual parameter op-timization has been used to optimize the valuesof the four parameters in the elongational viscos-ity model proposed by Sarkar and Gupta.9,10 Byminimizing the difference between the predictedentrance loss and the corresponding experimentalvalues at strategically selected flow rates, the opti-mization scheme starts by estimating the values ofelongation-thinning parameters, which is followedby estimation of the two elongation-thickening pa-rameters. The estimated elongational viscosities forDow 132i at two different temperatures are reported.The corresponding predictions of elongational

viscosity from Binding’s and Cogswell’s analysesare found to be in good agreement with the presentestimations.

Acknowledgment

The author thanks Prof. F. A. Morrison for makingthe experimental data for entrance pressure loss andshear viscosity available.

References

1. Bird, R. B.; Armstrong, R. C.; Hassager, O. Dynamics of Poly-meric Liquids; Wiley: New York, 1987; Vols. 1 and 2.

2. Macosko, C. W. Rheology Principles, Measurements andApplications; VCH: New York, 1994.

3. Meissner, J. Rheol Acta 1969, 8, 78.4. Cogswell, F. N. Polym Eng Sci 1972, 12, 64.5. Binding, D. M. J Non-Newtonian Fluid Mech 1988, 27, 193.6. Gupta, M. Polym Eng Sci 2000, 40, 23.7. Gupta, M. J Reinforced Plast Composites 2001, 20, 341.8. Gupta, M. SPE ANTEC Tech Papers 1999, 45, 83.9. Sarkar, D.; Gupta, M. J Reinforced Plast Composites 2001, 20,

1473.10. Sarkar, D.; Gupta, M. In CAE and Related Innovations for

Polymer Processing; Turng, L. S.; Wang, H. P.; Ramani, K.;Bernard, A. (Eds.); ASME: New York, 2000; MD-Vol. 90,p. 309.

11. Press, W. H.; Flanery, B. P.; Teukolsky, S. A.; Vetterling, W. T.Numerical Recipes; Cambridge University Press: Cambridge,1989.

12. Gupta, M. SPE ANTEC Tech Papers 2001, 47, 794.13. Arnold, D. N.; Brezzi, F.; Fortin, M. Calcolo 1984, 21, 337.14. Coupez, T.; Marie, S. Int J Supercomputer Appl High Perfor-

mance Computing 1997, 11, 277.15. Bathe, K. J. Finite Element Procedures; Prentice Hall:

Englewood Cliffs, NJ, 1996.


Estimation of elongational viscosity of polymers from entrance loss data using individual parameter...

Documents

Transcript of Estimation of elongational viscosity of polymers from entrance loss data using individual parameter...