2-d numerical simulation of differential viscoelastic fluids in a … · 2017. 2. 23. ·...

2-D Numerical Simulation ofDifferential Viscoelastic Fluidsin a Single-Screw ContinuousMixer: Application ofViscoelastic Finite ElementMethods

ROBIN K. CONNELLY, JOZEF L. KOKINIDepartment of Food Science, Center for Advanced Food Technology, Cook College, Rutgers University,New Brunswick, New Jersey 08901-8520

Received: July 11, 2001Accepted: October 8, 2002

ABSTRACT: Viscoelastic effects on mixing flows obtained with kneadingpaddles in a single-screw, continuous mixer were explored using 2-D finiteelement method numerical simulations. The single-mode Phan–Thien Tannernonlinear, viscoelastic fluid model was used with parameters for a dough-likematerial. The viscoelastic limits of the simulations were found using elasticviscous stress splitting, 4 × 4 sub-elements for stress, streamline upwind, andstreamline upwind Petrov–Galerkin (SUPG). Mesh refinement and comparisonbetween methods was also done. The single-screw mixer was modeled by takingthe kneading paddle as the point of reference, fixing the mesh in time. Rigid

Correspondence to: Jozef L. Kokini; e-mail: [email protected].

Contract grant sponsor: USDA National Research InitiativeCompetitive Grants Program.

Contract grant number: 2000-1768.

Advances in Polymer Technology, Vol. 22, No. 1, 22–41 (2003)C© 2003 Wiley Periodicals, Inc.

2-D NUMERICAL SIMULATION OF VISCOELASTIC FLUIDS IN A SINGLE-SCREW MIXER

rotation and no slip boundary conditions at the walls were used with inertiataken into account. Results include velocity, pressure, and stress profiles. Theaddition of viscoelasticity caused the shear and normal stresses to vary greatlyfrom the viscous results, with a resulting loss of symmetry in the velocity andpressure profiles in the flow region. C© 2003 Wiley Periodicals, Inc. Adv PolymTechn 22: 22–41, 2003; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/adv.10038

KEY WORDS: FEM, Mixing, Modeling, Simulation, Viscoelastic properties

Introduction

U se of high shear, single or twin blade, batch orcontinuous mixers for the processing of highly

viscous materials is an important unit operation inthe production of a wide variety of products, includ-ing adhesives, doughs, paints and coatings, candy,resins and rubbers, pet foods, pharmaceutical prod-ucts, soaps, etc. Successful mathematical modelingand computer simulation of the flow and mixing ina mixing operation would be a cost effective wayto obtain quantitative mixing parameters. It is farless expensive than machining and building mixergeometries and testing them for mixing ability. Itis also one of the most effective ways to gain in-sights about the operation of a system where ade-quate physical measurements to characterize the sys-tem are not readily possible, such as in an opaquedough, paint, gum, resin, or rubber. One can per-ceive mathematical simulation as an effective wayto nonintrusively probe a process and learn aboutwhat goes on inside the material being mixed. How-ever, modeling of mixing flows in realistic mixers,especially with viscoelastic materials, such as doughor synthetic polymers, is challenging due to the com-plex geometry and rheology. These geometries mayalso lead to many singularities that cause the mod-eling methods to break down with viscoelastic fluidmodels. In fact, use of viscoelastic fluid models inmixing simulations is rare. Finite element method(FEM) modeling of a simple paddle impeller in acylindrical vessel for an Oldroyd-B fluid was doneby Ann-Archard and Boisson1 using the Galerkinmethod and 4 × 4 Streamline upwind (SU) and com-pared to similar experimental velocity data.2,3 The2-D assumption in this case was found to give simu-lation results that agreed well with the experimentalresults. Petera and Nassehi4 used the Maxwell andthe Phan–Thien Tanner (PTT) models in a simula-tion of a free surface viscoelastic flow for rubber in

an internal mixer. The modeling was done usingGalerkin FEM with SU Petrov–Galerkin (SUPG). Thefree surface was handled by a modified pseudoden-sity method that eliminated the convection terms inthe free surface model equations and thus the needfor artificial physical parameters. The results werevalidated with a novel experimental rig.5 Boundaryelement analysis of a simple mixer with a Maxwelllinear viscoelastic fluid model found the viscoelas-ticity to only influence the evolution of stress atthe rotor and walls, but not the velocity profile.6

Dhanasekharan and Kokini7 modeled the 3-D flowof a single mode PTT fluid in the metering zone of acompletely filled single-screw extruder. It was mod-eled by means of a stationary screw and rotating bar-rel. The leakage flow between the screw and barrelwas neglected, leading to minimal effects due to vis-coelasticity being observed.

There are many computational fluid dynamics(CFD) approaches to discretizing the equations ofconservation of momentum, mass, and energy to-gether with the constitutive equation that defines therheology of the fluid being modeled and the bound-ary and initial conditions that govern the flow be-havior in a particular mixer. The most importantof these are finite difference (FDM), finite volume(FVM), and finite element (FEM). Others such asspectral schemes, boundary element methods, andcellular automata are used in CFD but their use islimited to special classes of problems.8 The modelingof viscoelastic fluid flows has been most successfulwith FEM. During early attempts at modeling vis-coelastic fluid flow with FEM, the solutions invari-ably became unstable at unrealistically low valuesof the Weissenberg number (Wi) or the related Deb-orah number (De).9 This problem, called the “highWeissenberg number problem,” is for the most partdue to the hyperbolic character of the advective termin the viscoelastic constitutive equations, which in-creases in importance with increasing De.10 This is es-pecially apparent at geometrical singularities such as

ADVANCES IN POLYMER TECHNOLOGY 23


sudden contractions, sudden changes in boundaryconditions as represented by the stick-slip problem,and flow around a sphere or cylinder, all of whichare considered benchmark problems in viscoelasticflow simulation.11 At singularities, which are foundin most practical problems including the flow in mix-ers, there are stress concentrations that increase to in-finity as the mesh is refined. In addition, the presenceof normal stresses gives rise to very thin boundarylayers in complex flows. The high stress gradients inthese layers are believed to be the cause of problemsfor most numerical methods.10

In order to ease the problems caused by the highstress gradients, Marchal and Crochet12 introducedthe use of 4 × 4 subelements for the stresses, whichwas later proven to be optimal.13 These bilinear sub-elements smoothed the mixed method solution ofthe Newtonian stick-slip problem, as well as aidedin the convergence of the viscoelastic problem. In or-der to deal with the hyperbolic nature of the advec-tive term of the constitutive equation, Marchal andCrochet12 implemented two different modificationsto the standard mixed Galerkin method: the consis-tent SUPG formulation14 and the nonconsistent SUmethod. The basic idea of SU is to add artificial dif-fusivity (or viscosity) that acts only in the flow di-rection. Extended to a Petrov–Galerkin formulation,the standard Galerkin weighting functions are mod-ified to weigh the element upwind of the node moreheavily than the downwind element. Because the ar-tificial diffusivity is applied only in the flow direc-tion, crosswind distortions are eliminated. When theweighting is consistent across all terms, there is nooverall artificial diffusivity effect.

However, when originally applied to the stick-slipproblem by Marchal and Crochet,12 SUPG still failedwhen the constitutive equations were coupled withthe equations of motion and the incompressibilityconstraint. Therefore, they applied the artificial dif-fusivity only to the advective term of the constitu-tive equation. This nonconsistent application of theSU method, when combined with the use of 4 × 4subelements for stress, had great success, reachingDe numbers as high as 64 for a circular contraction.Unfortunately, the nonconsistent SU does give rise toartificial stress diffusion along the streamlines. Therehas been some question of the importance of the arti-ficial stress diffusion along the streamlines, with Luoand Tanner15 claiming it actually changed the natureof the constitutive equation and affected the results atany level of mesh refinement. However, while the er-ror of the stress term is of O(h) or worse16,17 where his on the order of the mesh size, advection dominates

diffusion as the mesh size decreases so that the artifi-cial diffusivity term becomes negligible.18 Therefore,the accuracy of problems solved using the SU 4 × 4method must be verified on at least two, preferablythree, finite element meshes of decreasing elementsize.17,19 Further work with SUPG showed that whileit is not as robust where there are stress singularities,it is more accurate (to O(h3)), especially for unequalelement sizes, and converges faster than SU.16,17

The success of the artificial diffusivity concept todeal with the “high Weissenberg problem” led toother methods as well. One of the most successfulmethods is known as elastic viscous stress splitting(EVSS). This is done by splitting the stress tensor intothe sum of the viscoelastic and Newtonian contribu-tions, which regularizes the behavior of the constitu-tive equations. The Newtonian contribution is givenby 2ηD, where D is the strain rate tensor and η is theNewtonian (or solvent) viscosity. D is considered asan additional unknown that is approximated with aleast square method.20–22

In light of these advances in using differential vis-coelastic models in FEM simulations and the lackof published examples using this technology withhighly viscous, viscoelastic materials in mixers, theapplication of these techniques to the developmentof flow profiles for a dough-like material in a 2-Dsimplified mixer that can be used to mix dough isexplored.

Materials and Methods

PROBLEM DESCRIPTIONAND MESH DEVELOPMENT

The 2-D geometry for a single-kneading paddlein a 2′′ circular barrel with complete fill is shown inFig. 1. Flow occurs in the shaded area between thebarrel and the paddle. The paddle shape was cal-culated using the equations of Booy23 for a 2′′, duallobed, self-wiping, twin-screw mixer, with α and βas shown in Fig. 1 . The paddles used were actually ofthe 2′′ Continuous Processor of Readco Manufactur-ing (York, PA), which is a low pressure, corotating,twin-screw mixer used in such diverse applica-tions as dough mixing, polymeric condensationand other chemical reactions involving polymers,incorporation of light weight powders into highlyviscous materials such as bulk molding compoundsand rubbers, chocolate conching, production ofpharmaceutical pastes for pill formulation, etc. The

24 VOL. 22, NO. 1


α/2

β/2Rs

R β

Rα

Rp

hθ

PaddleBarrel

Fluid

FIGURE 1. Geometry used in the simulations23 whereα = 4.7◦ (0.082 rad.), β = 9.6◦ (0.167 rad.), Rs = 2.54 cm(barrel radius), Rβ = 1.269 cm, Rα = 2.4895 cm, andCL = 3.809 cm (twin shaft clearance). For (β/2) ≤ θ ≤(90 − α/2), Rp = Rs − hθ (paddle radius) and hθ =Rs(1 + cos(θ − β/2)) − (CL

2 − Rs2 sin2(θ − β/2))0.5.

geometry seen here is similar to that seen in thekneading zone of a single-screw extruder. However,since the velocity in the z direction is not taken intoaccount and the pressure is low, this simulationmore realistically models a batch mixer consistingof a paddle in a cylindrical tank.3 To fix the mesh intime, the paddle is taken as the frame of referencewith the barrel rotating in a counterclockwisedirection. For a cylindrical system, Rauwendaalet al.24 found the pressure and velocity gradientsto be the same for either the barrel rotating or thepaddle rotating, except for the effect of centrifugaland Coriolis forces on the boundary conditions. Inorder to take into account centrifugal and Coriolisforces, rigid rotation of the entire system is assumedat a speed of −�, which is in a clockwise direction.The Coriolis force, which is a deflecting force due toflows in both the radial and angular directions, andthe centrifugal force, which is in the radial directionresulting from the angular velocity, are pseudoforcesdue to being in a noninertial reference frame. Ameasure of the importance of the Coriolis forcesis the Rossby number, which is of the order of theratio of the blade radius to the channel height in anextruder. If the Rossby number 1, Coriolis forces

are of negligible effect, while at around unity theyare of mixed effect, and are dominate at �1.25 Usinga generalized Newtonian fluid in a single-screwextruder Spalding et al.26 found centrifugal andCoriolis forces with a Rossby number of ∼10 to benegligible. In our case it varies from a minimum of1 in the middle of the channel to a value of 100 inthe gap, and therefore the Coriolis force may havesome non-negligible effects. Simulations at 1 and100 rpm without rigid rotation were done in orderto explore the importance of these pseudoforcesin this geometry. Inertia was taken into accountwith density set at 1.204 gm/cm3, which is thedensity of 43.2% moisture flour–water dough. Thetangential velocity of the barrel was set to �Rs. Noslip conditions were used at the barrel and paddlewalls.

The meshes used in this work were created usingGambit (Fluent). The 1480 element mesh picturedin Fig. 2 is representative of the meshes. The 360 ele-ment mesh was designed to be coarse with increased

FIGURE 2. An example of the meshes used in thesimulations. It is the 1480 element mesh.



refinement in areas of high gradients as indicatedby experience. In this case it is at the wall, in thegap, and near the entrance to the gap at the bladetip. The ratios used in the mesh spacing were cho-sen to meet this goal while at the same time creatingquadrilateral elements as near to square as possibleas indicated by the mesh tests described later. Ideally,refinement is done by dividing elements, but keep-ing the original nodes in the same place. The jumpfrom 360 to 600 doubled the elements on the sidesof the paddles. After that the meshes were refinedby adding elements as well as by setting the spac-ing ratios such that the element shape is optimizedaccording to the mesh tests. Direct doubling quicklymakes the number of elements enormous and theelement shape deteriorates.

The 360 and 600 element meshes have 5 elementsacross the channel, including across the gap, whilethe 1480 and 2080 have 10. The 3360 element meshhas 12. A ratio in spacing between nodes of 1.2 (360and 600) or 1.1 (rest) from walls to the center of chan-nel was used (e.g., if the spacing between the wallnode and the 2nd node is 1, then the spacing be-tween the 2nd node and the 3rd node will be 1.1,between 3rd and 4th 1.21, etc.) The 360 and 600 el-ement meshes have 26 and 50 elements across thepaddle sides and 10 across the paddle tip with a 1.2ratio from the paddle tip. The paddle sides for the1480 element mesh has 60 elements (1.1 ratio fromtip) while the 2080 element mesh has 90, and the 3360element mesh has 120 with a ratio from the paddletip of 1.05 on the wall and 1.1 on the paddle. Thepaddle tips contain 14 elements (1480 and 2080) or20 (3360) with ratio of 1.1 from the edge to the centerthe center of the tip.

Gambit offers a variety of mesh tests to evalu-ate the quality of meshes for use in FEM. Ideally,

TABLE IResults of Mesh Tests

Elements in the Mesh

Test 360 600 1480 2080 3360

Equiangle skew<0.1 35.6% 32.7% 46.2% 56.1% 54.9%<0.25 91.1% 87.3% 94.0% 88.2% 86.6%<0.5 100% 100% 100% 100% 100%

Stretch<0.25 1.1% 40.7% 14.9% 49.8% 43.6%>0.5 51.1% 12.0% 50.3% 22.1% 16.4%

perfectly square (or perfectly triangular) 2-D ele-ments are needed for the most effective use of FEM.However, realistic geometries do not lend them-selves to the use of only perfect elements. One ofthe goals in mesh creation is to keep the shape of theelements as close to ideal as possible, while still accu-rately representing the real geometry. These qualitytests are measures of how well a given mesh has suc-ceeded in meeting this goal. Equiangle skew, whichlooks at how close the angles of each mesh quadri-lateral are to 90◦, is defined as

QEAS = max{

�max − 9090

,90 − �min

90

}

where �max and �min are the largest and smallest anglerespectively in degrees. Stretch ratio, which looks athow close each mesh quadrilateral is to an equilateralelement, is defined as

Qs = 1 −√

2[min(s1, s2, s3, s4)]2

max[d1, d2]2

where si indicates side and di indicates diagonal. Forboth of these tests, a value of 0 indicates a perfectelement while a value of 1 indicates a degenerateelement. They combine to give a good overall pictureof the quality of the meshes in this work.27

As shown in Table I, all mesh quadrilaterals fallinto the excellent (<0.25) or good (<0.5) range forequiangle skew, although only the 2080 elementmesh is considered a high quality mesh with an av-erage value <0.1. The 360 element and 1480 elementmeshes score only fair for stretch ratio, while the 2080element mesh again scores the best. Unfortunately,the elements that scored only good according to the

26 VOL. 22, NO. 1


mesh tests occur near the blade tips, where instabil-ity will occur due to the high gradients at the walland the blade surface, as well as the stress concen-tration at the corner. The 2080 element mesh had thebest-shaped elements in these regions, and the resultis a more stable solution.

FLUID MODEL AND FLOW PARAMETERS

A typical application of this type of mixer is theprocessing of dough. Therefore, the PTT nonlinearviscoelastic model28 was chosen for its ability tomodel the behavior of a model flour–water dough aswell as its relatively good behavior in FEM simula-tion. The PTT constitutive model takes the followingform

exp[ελ

η1tr(T1)

]T1 +λ

[(1 − �

2

) ∇T1 + �

2

T1

]= 2η1D

The parameters are based on those of Dhanasakaranet al.29 as follows: the partial viscosity η1 = 88888.9,poise ε = 0.08, and � = 0.01. The relaxation time λ isevolved from a value of zero (the Newtonian case)to the point where the simulation diverges and will

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1.E+08

1.E-03 1.E-02 1.E-01 1.E+00 1.E+01 1.E+02 1.E+03

Shear Rate (1/s)

Ste

ady

Sh

ear

Vis

cosi

ty (

po

ise)

.

NewtonianPTT 1 mode/l=0.5 sPTT 1 mode/l=10 sPTT 1 mode/l=100 sPTT 1 mode/l=1000 sPTT 4 mode dough

FIGURE 3. Steady shear profiles for single mode PTT models used in the 2-D simulations with the four mode PTTdough model of Dhanasekharan et al.29 as a reference.

no longer give a solution. The stress was consid-ered to be divided with a purely viscous componentT2 = 2η2D, such that the total stress T = T1 + T2with D being the rate of deformation. The result-ing viscosity ratio was set to η1/(η1 + η2) = 1/9. Thesteady shear profiles for several of the values of therelaxation time used in this work are shown in Fig. 3.The shear thinning behavior of these models is ex-hibited only in a very limited range of shear rates thatrange from around 2–200 s−1 for λ = 0.5 s to about0.001–0.1 s−1 for λ = 1000 s. We shall see later thatthe shear rate range observed in the mixer at 1 rpmwas from around 0.1 to 30 s−1, which is in the shearthinning range of most of the models considered.Figure 3 also shows that the 4 mode dough model ofDhanesekaran et al.29 is very similar in viscosity tothe single mode PTT models used in this work in theshear rate range of interest, especially at the lowerrelaxation times.

Typically in a mixer, the Deborah Number, whichis defined as the ratio of the characteristic materialtime to the characteristic process time, is taken tobe De = λ� and is the definition we use here.30 TheWeissenburg number, which is defined as the ratioof the elastic force to the viscous force, is described



as Wi = (1/η)�, where 1 is the primary normalstress coefficient and η is the viscosity.30 The primarynormal stress coefficient is defined as

1 ≡ −(τ11 − τ22)�̇ 2 = N1

�̇ 2

under steady simple shear flow conditions. It isrelated empirically to the shear rate (�̇ ) by theexpression30 1 = m′|�̇ |n′−2. In the case of the PTTmodel, η and 1 are shear rate dependant, so Widoes not have a global meaning but only a local. Theprimary normal stress coefficient (1) for the PTTmodel in viscometric flow28 is given by

1 = 2η1λ

1 + �(2 − �)λ2�̇ 2

The viscometric flow relations will be used as esti-mates of the actual values during mixing and simu-lation.

SIMULATION METHODOLOGY

For flow problems of viscoelastic fluids of thedifferential type, Polyflow first develops a mixedGalerkin formulation of the governing equations.For this steady-state, 2-D isothermal flow wherethe domain χ is the space between the paddle andthe barrel, the viscoelastic extra-stress, velocity, andpressure are approximated respectively by means ofthe finite expansions:

T a1 =

NT∑i=1

T i1φi va =

Nv∑j=1

v jψ j pa =Np∑

k=1

pkπk

where φi , ψ j , and πk represent given finite elementbasis functions, while T i

1, v j , and pk-are unknownnodal values. Substituting the approximation intothe isothermal governing equations, using the mixedGalerkin method, the following set of equations isobtained:

∫χ

φi

[exp

[ελ

η1tr(T1)

]T a

1 + λδT a

1

δt− 2η1Da

]dχ = 0

(constitutive equation)

∫χ

{ψ jρ

[dva

dt− f

]+ ∇ψT

j ·[−pa I + 2η2Da + T a1

]}dχ

=∫∂χ

ψ j � · n ds (conservation of momentum)

∫χ

πk[∇ · va ] dχ = 0 (conservation of mass)

where δTa1

δt = �2

T1 + (1 − �2 )

∇T1 and f contains the Cori-

olis and centrifugal terms. Whenλ vanishes, the con-stitutive equation reduces to the generalized Newto-nian case.31 Because of the hyperbolic nature of theλv · ∇T1 term from the upper- and lower-convected

derivatives of the extra-stress∇T1, the accuracy and

stability of the mixed Galerkin formulation deterio-rates as the elasticity number increases in flows withboundary layers or singularities.10 The numerical re-sults can then be stabilized by the use of viscoelasticextra-stress interpolations such as SU or SUPG. Themost robust technique, SU, is done by applying anartificial diffusivity K = k̄ vv

v·v to the hyperbolic termonly in the stream-wise direction. The discrete con-stitutive equation then becomes:

∫χ

φi

[A

(T a

1 , λ) · T a

1 + λ(�̇ )δT a

1

δt− 2η1Da

]dχ

+∫χ

λk̄va

va · va

(va · ∇T a

1

) · ∇φi dχ = 0

where k̄ is a scalar on the order of the mesh size.12

While this technique gives rise to artificial extra-stress diffusion along the streamlines, the impor-tance of stress diffusivity decreases when the finiteelement mesh is refined.17 Therefore, five meshes ofdecreasing size were used to verify the results. Incontrast, SUPG applies K to all the terms in the dis-crete constitutive equation, eliminating the artificialdiffusion.14

The viscoelastic extra-stress field interpolationtechniques used in this work are elastic-viscousstress splitting (EVSS) and 4 × 4 bilinear subele-ments for the stress. In EVSS, the stress tensor issplit into elastic and viscous components. When theT = T1 + T2 form of the stress tensor is used, the con-vected derivative of the rate of strain emerges, whichrequires a second-order derivative of the velocityfield. To overcome this, D is considered an unknown

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-5000

0

5000

10000

15000

20000

25000

30000

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

X Coordinate (cm)

τ 22

(g/c

m2 )

3360 elements2080 elements1480 elements360 elements

FIGURE 4. Normal stress τ22 during mesh refinement for the PTT Model with λ = 100 s on a circle of radius 2.489 cm.

and is obtained by a least squares approximationusing the definition D ≡ 1

2 [(∇v) + (∇v)T ].20,22 Themodified discrete constitutive equation and strainrate equation are as follows:

∫χ

φi

[A(T a , λ) · T a + λ(�̇ )

[δT a

δt+ 2η1

∇Da

]]dχ = 0

and∫χ

�i[Da − (∇va + ∇T va )/2

]dχ = 0

Alternatively, the solution can be stabilized by divid-ing each quadrilateral in the mesh into 16 bilinearsubelements for calculating the stress.12

An iterative technique called evolution was usedto gradually introduce the source of the instabilityinto the solution. For differential, viscoelastic prob-lems, evolution is usually done on either the relax-ation time (λ) or the mixing speed (�). This givesgradually increasing values of De. In this case ithas been chosen to evolve λ at a range of fixedmixing speeds that go from a very low value of 1rpm, which is in effect creeping flow, to a more typ-ical speed for the mixer on which this simulation isbased of 100 rpm. Then it is possible to see what

level of λ can be achieved at each speed, using theabove numerical techniques. Evolution is appliedto the relaxation time such that λi = f (Si ) × λmax,where f (Si ) = Si = (Si−1 + Si ) where S0 = 0 andSfinal = 1 and S0 = 0.0001. Then the solution is ob-tained using Newton’s iterative scheme with the con-verged solution of the previous step used as theinitial guess when available. If the solution con-verges Si = Si−1 × 1.5. If the solution diverges,Si = Si/2 and the iteration is re-done until Si is

TABLE IIAnalysis of Mesh Refinement Error of τ22 at 1 rpm withλ = 100 s

Elements in the Mesh

Test 360 1480 2080

Pearson correlation 0.9735 0.9908 0.9953coefficient (R)

Without tip elements 0.9950 0.9994 0.9997

R2 0.9477 0.9817 0.9906

Data read at points located on a circle of 2.489 cm are com-pared with linear estimates of data from the 3360 mesh at thesame points.



TABLE IIILimits of De Reached by Several Methods Used in Viscoelastic Simulations During Mesh Refinement at 1 rpm

EVSS SUPG 4 × 4 SUPG EVSS SU 4 × 4 SU

Mesh Size λ (1 rpm) De λ (1 rpm) De λ (1 rpm) De λ (1 rpm) De

360 elements 0.327 0.034 0.23 0.024 651.04 68.2 1000 104.7

600 elements 0.178 0.019 1.04 0.109 14.12 1.47 23.4 2.45

1480 elements 0.089 0.009 0.089 0.009 0.73 0.076 131.78 13.8

2080 elements – – 0.066 0.007 0.79 0.082 543.58 56.9

3360 elements – – – – 0.58 0.061 110.32 11.6

less than a minimum Smin = 1 × 10−5 and the sim-ulation stops.31

Results and Discussion

VISCOELASTIC LIMITS AND ACCURACYOF SIMULATION RESULTS

The coarsest mesh of 360 elements is somewhatinaccurate, but by the time the mesh size reached1480 elements, the discretization error is minimal.The main difference between meshes with the vis-coelastic fluids is the magnitudes of the pressure,shear stress, and, in particular, normal stresses calcu-lated at the corner singularity as shown in Fig. 4. Thevariation in the values of normal stress τ22 with a re-laxation time of 100 s at the blade surface is the worstcase of the results read on a circle of radius 2.489 cm.(Note that the data on this and subsequent graphsis read where a circle of the given radius crosses amesh element. The x-axis is the x-coordinate of thedata points with the origin the barrel center. Mostx values will generate two data points at the corre-sponding positive and negative y positions.) Statis-tical tests of the equality of linear estimates of 3360element mesh values with the actual values from the360, 1480, and 2080 element meshes are shown inTable II. The square of the Pearson correlation coef-ficient (R2) of all three meshes to the 3360 elementmesh is adequate. If the elements within 0.04 cm ofthe tip are disregarded, the results are nearly exact forthe 1480 and 2080 element mesh, indicating that ar-tificial stress diffusion is negligible throughout mostof the flow domain.

Coarser meshes allowed convergence at higherDe since effect of the discontinuity is smoothed.This mesh refinement effect has been noted by other

authors.22 The more accurate SUPG technique is notadequate for this geometry because of the cornersingularity at the blade tip, as is demonstrated inTable III. The less computationally intensive EVSStechnique is also not as effective as using 4× 4 bi-linear subelements for stress. Therefore, the 4× 4 SUtechnique was determined to be the method neces-sary to solve this problem. Even this technique wasunable to reach the desired relaxation time of 1000 sat low rpm values. The excellent quality of the 2080element mesh is demonstrated by the high relax-ation time it is able to achieve in relation to the othermeshes before the solution diverges. Instabilities atthe start of evolution with the higher speeds necessi-tated the use of different parameters for the evolutionprocedure. A Smin of 1 × 10−5 used 1 and 10 rpmwhile a Smin of 1 × 10−6 used 60 and 100 rpm.

At the high rpm values that are more represen-tative of the actual conditions found in this type ofmixer, the instabilities in the calculations and some ofthe effects of the viscoelasticity seem to disappear asseen in Table IV. At 1 and 10 rpm, the calculations arehighly unstable and able to achieve a maximum De ofonly 13.5. However, at 60 rpm, while there is someinstability in the calculations at the start of evolu-tion, the desired maximum relaxation time of 1000 s

TABLE IVLimits of De Reached at Several Blade Speeds Usingthe 1480 Element Mesh and 4 × 4 SU

Blade Speed λ De

1 rpm 131.78 13.8

10 rpm 12.8 13.4

60 rpm 1000 6283

100 rpm 1000 10472

1000 was the highest λ attempted.

30 VOL. 22, NO. 1


is achieved. The last evidence of instability in the 60and 100 rpm simulations occurs at De ≈ 13.5. Thisis at the same De at which the 1 rpm and 10 rpmsimulations broke down with this mesh. One pos-sible reason that the higher rpm simulations wereable to go beyond De = 13.5 is the high value of theshear rates throughout the flow at those velocities.The shear rate range seen at 100 rpm is on the orderof 10–1000 1/s as shown in Fig. 12. From the steadyshear profiles of the PTT constitutive models shownin Fig. 3, it becomes evident that the shear thinningregion would be of consequence with only the verysmallest values of the relaxation time, where in factthe simulation did show evidence of instability. Forthe higher values of the relaxation time, the behaviorof the Newtonian component of the viscosity dom-inates the simulation, thus stabilizing the solutionand allowing the high De values that were achieved.Another way to think about this is to note that whenthe De is very high, either the characteristic materialtime is long or the characteristic process time is shortor both. In either case, the material will not have timeto relax and will follow its unrelaxed behavior pro-file. In this case, that is the behavior of the Newtoniancomponent.

-2.00E+06

-1.50E+06

-1.00E+06

-5.00E+05

0.00E+00

5.00E+05

1.00E+06

1.50E+06

2.00E+06

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

X Coordinate (cm)

Pre

ssu

re (

po

ise)

l=0 sec.l=0.5 sec.l=10 sec.l=51 sec.l=100 sec.

FIGURE 5. Pressure profiles at 1 rpm on the 2.489 cm radius circle with the 1480 element mesh for the PTT at severalrelaxation times.

Another possible explanation for the behavior ofthe simulation at 100 rpm is that at higher velocities,the inertial forces come to dominate the elastic forcesgenerated in the flow. A measure of the importanceof elastic effects on mixing flow is the elasticity num-ber (El) defined as El = Wi/Re = 1/(ρD2) whereD is the blade diameter and ρ is the fluid density.When El > 0.25–0.5, elastic forces have been foundto dominate the inertial forces, causing completeflow reversal the secondary flow profile in somemixer geometries.30,32,34 In our case, when λ = 100 sthe El � 0.25 at 100 rpm in the high shear regionsand El 0.5 at 1 rpm throughout most of the flowdomain, indicating that elasticity dominates at thelow speed and inertia is important in high shear re-gions near the blade tip at the high speed. The effectof the inertia dominating the flow in the gap regionis to dampen the viscoelastic effects and thereforestabilize the solution.

The centrifugal and Coriolis forces, which wereincluded through use of rigid rotation, were a fac-tor at high values of the relaxation time. For thePTT model at 1 rpm and λ = 100 s, they changedthe stresses about 13%, the velocity magnitude byan average of 0.4%, and shear rate and pressure by



an average of 2–3%. At an equivalent De with λ = 1 sat 100 rpm they changed the solution only negligibly.No change in any variable was detected at 1 rpm insimulations using a Newtonian fluid or a shear thin-ning Bird–Carreau generalized Newtonian fluid. Aside benefit of including the Coriolis and centrifu-gal forces with the PTT model fluid was that theyhad a stabilizing effect on the solution. Without rigidrotation, the maximum De achieved was only 13.4at 100 rpm, while at 1 rpm it was only possibleto reach that De level by using the more effective,but more computationally expensive 2080 elementmesh.

VISCOELASTIC EFFECTS

The pressure variation profiles at 1 rpm for a se-ries of relaxation times read on a circle of radius 2.489cm are shown in Fig. 5, where the pressure valueof zero is equivalent to ambient. These profiles arerepresentative of the overall mapping of the pres-sure, since the pressure varies mainly with the an-gle from the center of rotation. It is interesting to

-2.00E+06

-1.50E+06

-1.00E+06

-5.00E+05

0.00E+00

5.00E+05

1.00E+06

1.50E+06

2.00E+06

-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5

X Coordinate (cm)

Pre

ssu

re (

dyn

e/cm

2 )

Newtonianl=60, n=0.5l=60,n=0.2

FIGURE 6. Pressure profiles at 1 rpm on the 2.489 cm radius circle for the Bird–Carreau viscous model(T1 = 2[η∞ + (η0 − η∞)(1 + λ2γ 2)(n−1)/2]D) fluids with η0 = 100000 and η1 = 11111.1 poise at several levels of shearthinning. Note that the steady shear profile of the Bird–Carreau model with λ = 60 and n = 0.2 is nearly identical to thatof the single mode PTT model used in this work with λ = 100.

note that at a relaxation time of 0 s where the PTTmodel is equivalent to the Newtonian model, thepressure on the front and back of the blade is equaland opposite. However, as the relaxation time in-creases, two changes are observed: the magnitude ofthe pressure drops and there is a loss of symmetry inthe profile. The change in magnitude was observedfor purely shear thinning materials as shown in Fig. 6for a series increasingly shear thinning Bird–Carreauviscous constitutive model fluids, but not the loss ofsymmetry between the front and back of the blades.Therefore, it appears that the loss of symmetry in thepressure differential between the front and back ofthe blades is a purely viscoelastic effect. Note alsothat at a relaxation time of 0.5 s, the asymmetry ofthe PTT fluid is of a different form from that seen atthe higher relaxation times.

Related to effect on the pressure profiles, a loss ofsymmetry with increasing viscoelasticity is also ev-ident in the velocity distributions. The velocity pro-file in the rotating reference frame is the secondaryprofile since the primary tangential components ofthe velocity were removed. It has been shown that

32 VOL. 22, NO. 1


FIG

UR

E7.

Vel

ocity

vect

ors

colo

red

acco

rdin

gto

mag

nitu

deat

1rp

min

the

mid

dle

ofth

eflo

wdo

mai

nfo

rre

laxa

tion

times

of(a

)0

s(N

ewto

nian

case

)an

d(b

)10

0s

whe

reth

eun

itsof

velo

city

are

cm/s

.



viscoelastic effects are more apparent in the sec-ondary profile.30 In this case, the velocity profile con-sists of a central point around which the bulk of fluidis circulating as shown in Fig. 7 and some fluid flow-ing through the gap, with dead zones just before andbehind the blade tip where the flow splits betweenthe main circulating flow and the flow through thegap as shown in Fig. 8. The asymmetry is evidentin the center of circulation, which moves toward thefront of the blade with increasing viscoelasticity asshown in Fig. 7b. Also, the blade tip dead zones

FIGURE 8. Velocity magnitude distribution at 1 rpm in the gap region for relaxation times of (a) 0 s (Newtonian case)and (b) 100 s where the units of velocity are cm/s.

are symmetric in the Newtonian case, but becomeincreasingly asymmetric with increased viscoelastic-ity as demonstrated in Fig. 8b. The velocity dividedby the time per revolution profile at x = 0 is shownin Fig. 9. Note that the most distortion of the profiledue to viscoelasticity is found with the 0.5 s relax-ation time material and gradually returns to a profilesimilar to the Newtonian material as the relaxationtime increases at 1 rpm. This is in spite of the factthat the pressure decreases substantially due to shearthinning, thus decreasing the pressure differential

34 VOL. 22, NO. 1


0

2

4

6

8

10

12

14

16

2.48 2.49 2.5 2.51 2.52 2.53 2.54

Y coordinate (cm)

Vel

oci

ty M

agn

itu

de

(cm

/rev

) .

r.t. = 0 sec.r.t. = 0.5 sec.r.t. = 10 sec.r.t. = 100 sec.10 rpm60 rpm100 rpm

FIGURE 9. Velocity (cm/s) divided by the blade speed (rev/s) across the gap on the X = 0 cm line at 1 rpm or at arelaxation time of 100 s unless otherwise noted.

-2.00E+08

-1.50E+08

-1.00E+08

-5.00E+07

0.00E+00

5.00E+07

1.00E+08

1.50E+08

2.00E+08

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 2.5

X Coordinate (cm)

Pre

ssu

re (

po

ise)

l=0(Newt.)

l=1 sec.

l=100 sec.

FIGURE 10. Pressure profiles at 100 rpm on the 2.489 cm radius circle with the 1480 element mesh for the PTT atseveral relaxation times.



resistance to the flow. The flow through a gap in onerevolution is 2.5–3% of the overall material, as calcu-lated by taking the sum over the gap of the averagevelocity between two points multiplied by the dis-tance between the points, dividing by the time perrevolution, and then comparing the result to the to-tal flow area. Viscoelastic effects appear to enhanceflow through the gap slightly in accordance with theamount of distortion seen in the velocity profile.

As the blade velocity increases, the velocity pro-file in the gap at a relaxation time of 100 s remains

FIGURE 11. Velocity magnitude distributions at 100 rpm in the gap region for relaxation times of (a) λ = 1 s and (b)λ = 100 s where the units of velocity are cm/s.

stable until 100 rpm, where inertia starts to be-come significant. Also, the effects of the viscoelas-ticity are less visible in the pressure and veloc-ity profiles at 100 rpm and λ = 100 s as shown inFigs. 10 and 11b, again indicating that the New-tonian component is dominating at this relaxationtime as discussed previously. In contrast, at λ =1 s the 100 rpm profiles (see Figs. 10 and 11a) aresimilar in shape to those at 1 rpm with λ = 100 s(see Figs. 5 and 8b), which is where the De isequivalent.

36 VOL. 22, NO. 1


0

2

4

6

8

10

12

14

16

18

20

2.48 2.49 2.5 2.51 2.52 2.53 2.54

Y position at X=0 (cm)

Sh

ear

Rat

e at

1 r

pm

(1/

s)

0

100

200

300

400

500

600

700

800

900

Sh

ear

rate

at

100

rpm

(1/

s)

1 rpm, Newt.1 rpm, l=0.5 s1 rpm, l=10 s1 rpm, l=100 s100 rpm, Newt.100 rpm, l=1 s100 rpm, l=100 s

a

0

5

10

15

20

25

30

35

40

0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25

X Coordinate of 2.489 cm radius circle (cm)

Sh

ear

Rat

e at

1 r

pm

(1/

s)

0

100

200

300

400

500

600

700

800

900

1000

Sh

ear

Rat

e at

100

rp

m (

1/s)

1 rpm, Newt.1 rpm, l=0.5 s1 rpm, l=10 s1 rpm, l=100 s100 rpm, Newt.100 rpm, l=1 s100 rpm, l=100 s

b

FIGURE 12. Effect of relaxation time of the PTT model on the shear rate (a) in the gap at the x = 0 line and (b) near theblade tips on the 2.489 cm radius.



FIGURE 13. Shear stress distributions at 1 rpm for relaxation times of (a) 0 s (Newtonian case) and (b) 100 s where theunits of stress are gm/cm2.

FIGURE 14. Normal stress distributions at 1 rpm for a Newtonian fluid (λ = 0 s), (a) τ11 and (b) τ22 where the units ofstress are gm/cm2.

38 VOL. 22, NO. 1


FIGURE 15. Normal stress distributions at 1 rpm for a relaxation time of λ = 100 s, (a) τ11 and (b) τ22 where the units ofstress are gm/cm2.

The shear rate in the flow domain is low, exceptnear the blade tip and in the gap. Shear rates in thegap and approaching the front and backsides of theon a circle of 2.489 cm are shown in Fig. 12. Fromthe steady shear profiles in Fig. 3, it is seen that forλ = 0.5, 10 and 51 s, the models are in the shearthinning regions at 1 rpm. In contrast, the Newto-nian λ = 0 has a constant viscosity of η0 and whenλ = 100 s the shear rate is such that a constant vis-cosity of η∞ has been achieved in the gap. This isreflected in the way the shear rate varies in the gapin Fig. 12a, with the constant viscosity fluids linearand the shear thinning fluids curved. The slightlyviscoelastic λ = 0.5 s fluid has a much higher shearrate at the tip than the Newtonian or long relaxationtime fluids, reflecting the differences in the pressureprofile at this relaxation time noted earlier. The vis-coelastic fluids have asymmetry in their shear rateprofiles also, with higher shear rates at the front ofthe blade seen in Fig. 12b.

Changes in the stress distributions with the ad-dition of viscoelasticity are dramatic as shown inFigs. 13–15. For the Newtonian case shown in Fig. 13,the normal stresses are not zero since this is not a vis-cometric flow, but they are equal and opposite suchthat the first normal stress difference N1 = |τ11| −|τ22| ≈ 0. The first normal stress difference varies

from zero near the corner singularity because of theerror in the SU method. This error is not found fora Newtonian fluid solved using Galerkin’s methodwithout the use of SU. For the viscoelastic case, theexpected power law relationship for 1 is seen asshown in Fig. 16. Comparisons of the simulation re-sults with the PTT steady shear expression for thefirst normal stress coefficient calculated at the sameshear rates show that the simulation results are qual-itatively in agreement with the calculated results, es-pecially when taking into account that this is not aviscometric flow simulation. The shear rates usedare the second invariants of the rate of deformationtensors calculated in the simulation. At the highershear rates that are found near the corner singular-ity, some deviation in the simulation data is apparentin the graph, which again is thought to be due to theerror inherent in the SU method.

Conclusions

The process of solving a flow problem in a mixerfor a differential viscoelastic fluid, using FEM is laidout, with a thorough look at the issues involved in



y = 32151.06x-1.73

R2 = 0.99

y = 88697.51x-1.99

R2 = 1.00

100

1000

10000

100000

0.1 10 100

Shear Rate (1/s)

Ψ1

(g/c

m)

Data

PTT

Data Fit

PTT Fit

1

FIGURE 16. Primary normal stress difference (N1) vs. shear rate (γ̇ ) at 1 rpm for a relaxation time of λ = 100 s.

developing a mesh, choosing an appropriate tech-nique to handle the instabilities inherent in differen-tial viscoelastic fluid models, understanding the er-ror in the solution, and analysis of the data in termsthat apply to mixers. Of the techniques available tous for the handling of differential viscoelastic fluidmodels, only the SU with 4 × 4 subelements for stresstechnique of Marchal and Crochet12 was able to solvethis problem with a level of viscoelasticity useful forthe study of dough mixing. While the error of theSU method is evident near the corner singularity inall the meshes studied, throughout most of the flowdomain the mesh discretization error is of negligibleimportance so that qualitatively all the meshes givesimilar information. The effect of increasing rpm isto increase the magnitude of all the variables includ-ing the shear rate, and therefore shift the importantviscoelastic effects to lower relaxation times. The im-portance of inertial forces also increases at higherspeeds and thus stabilizes the solution. Centrifugaland Coriolis forces were found to generally stabilizethe solution of the viscoelastic problem, with non-negligible effects on all variables at higher values ofthe relaxation time. In contrast, they had no mea-surable effect on simulations with purely viscousmodels. The general effect of viscoelasticity shown

in this work is to introduce asymmetry into the pres-sure and flow profiles, as well as greatly modifyingthe stress profiles from those for the Newtonian case.There are also reductions in the magnitudes of thepressure and stresses that are due to shear thinningeffects.

The implication of these results is that a usefulsimplifying approach to modeling a viscous, vis-coelastic materials such as dough or synthetic poly-mers that have a spectrum of relaxation times andviscosities is to tailor the viscoelastic models to in-clude only those parameters that affect the rheol-ogy in the shear rate range of interest. For example,since only relaxation times up to around 2 s are shearthinning in the shear rate range generated in this ge-ometry at 100 rpm, higher relaxation times can bedropped since they would not be able to relax at thischaracteristic process time and therefore are not con-tributing to the viscoelastic or shear thinning effects.Also, the stabilizing Newtonian component wouldneed only be set to a value that would take over atshear rates higher than those seen in the process ofinterest, rather than the actual infinite shear viscos-ity seen in experiments. These simplifications allowthe challenges of FEM with differential viscoelas-tic fluid models to be overcome, while allowing the

40 VOL. 22, NO. 1


exploration of the effects of real fluids in a realisticgeometry.

Acknowledgment

This is publication No. D10544-14-01 of the NewJersey Agricultural Experiment Station supported bythe Center for Advanced Food Technology (CAFT).The Center for Advanced Food Technology is aNew Jersey Commission on Science and TechnologyCenter.

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