the finite element method for flow and heat transfer analysis

The Finite Element Method for Flow and

Heat Transfer Analysis*

Evan Mitsoulis and John Vlachopoulos Department of Chemical Engineering

McMaster University Hamilton, Ontario, Canada

ABSTRACT

The finite element method (FEM) is discussed and a specific formulation for flow problems is outlined that can encompass non-Newtonian inelastic and viscoehtic fluids. A temperature for-

mulation is also considered t h t can be applied for nonisothermal analyses of fluid flow. Some illus- trative examples of the application of the method in polymer processing are also presented.

INTRODUCTION

he finite element method (FEM) is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineer-

ing problems (Huebner & Thornton, 1982). Like the better known finite difference method (FDM), the FEM is used to solve the appropriate differential equations that describe these problems. The term “finite element analysis” is sometimes used incorrectly to describe macroscopic mass and energy balances performed for a specified number

T

*Financial assistance from the Natural Sciences and Engineering Research Council of Canada is gratefully acknowledged.

ADVANCES IN POLYMER TECHNOLOGY

of control volumes (i.e., “elements” of the total volume). In fact there exist several commercially available computer packages for injection mold design with such misleading labels. This is an unfortunate situation and it is time that the differ- ences between these methods are clearly under- stood by everyone in the field of polymer technology. The term “finite element” should be used to describe numerical techniques based on variational or weighted residual principles (Schechter, 1967; Finlayson, 1972).

The FEM was originally developed for stress analysis in complex airframe structures during the 1960s. Because of its diversity and flexibility as an analysis tool, it has since been extended and applied to the broad field of continuum mechanics.

107

FINITE ELEMENT METHOD FOR FLOW AND HEA T TRANSFER ANALYSIS

Zienkiewicz and Cheung (1965) showed that the method is applicable to all field problems that can be cast into variational form. Some field problems that have been solved with the FEM include elec- trostatic fields, magnetostatics, seepage, torsion, irrotational flow, heat conduction, electric conduction, diffusion flow in porous media, etc. Fur- thermore, Oden (1969a and 1969b) gave a generalized interpretation of finite element models and showed how finite element equations can be developed from well-established global energy balances without resorting to classical variational principles. All these contributions served to expand the use of FEM for virtually any problem that can be described by differential equations.

Since the early 1970s, the FEM has been applied for the solution of fluid mechanics problems and, in particular, to slow viscous flows (small Rey- nolds number) that are usually encountered in the processing of polymer melts (Palit & Fenner, 1972; Taylor & Hood, 1973; Tanner, 1973; Nickell, Tanner, & Caswell, 1974). Recent at- tempts to use finite elements for the study of flow of viscoelastic fluids are summarized and critically examined by Crochet and Walters (1983) and Mitsoulis, Vlachopoulos, and Mirza (1983a).

The main advantage of FEM over other numerical methods (notably, finite difference) is its ability to solve problems in irregular and complex geometries with unusual boundary conditions (Vlachopoulos, 1977). Once the general differential conservation equations have been cast in their equivalent finite element formulation, a computer program can be written and used for different situations with only minor changes. These may include the geometry, boundary conditions and material properties that are particular to each problem.

The main disadvantage of the method is its complicated formulation that requires a good understanding of variational or weighted residual principles and a good command of matrix algebra and computer programming. However, the development of user-friendly computer packages can help people unfamiliar with the intricacies of the method, to apply a powerful tool to different engineering problems and obtain useful solutions.

HOW THE METHOD WORKS

In this section we will focus our attention on fluid flow, although the analysis is quite general

108

and can be applied to other fields as well. Fluids flow within specified solid boundaries

(e.g., process equipment) or flow freely in the atmosphere in unconstrained flows with free surfaces, The flow field can be described by the well- known conservation equations of mass, momentum, and energy. For two-dimensional planar do- mains and creeping flow (Re << l), these equations take the form

while at the boundary surface we have

Tx = u,,i + 7.,j ( 4 4

T, = + uY j (4b)

q = q,i + q,j (44

(I.. = -p&. + 7..

-

with

( 5 ) 11 11

where

(I,,, a,,, 7,,, 7yy, 7xy = stress components bx,b, - -

TMT,

= body forces acting on a plane area A (see Fig. 1)

= applied surface tractions on some part r, of the contour r enclosing A

T = temperature qx,qy

VXYYY = velocity components i ,j

= applied heat input per unit area on some part rs of r

= direction cosines of the unit outward normal vector n to the boundary r at any point

6ij = Kronecker delta.

In the finite difference method (FDM) we proceed by writing the above equations as difference equations for an array of grid points. Therefore in the FDM we have a “pointwise” approximation to the governing equation. In the FEM we divide the domain A in subdomains (finite elements) (see

VOL. 4, NO. 2

FINITE ELEMENT METHOD FOR FLOW AND HEAT TRANSFER ANALYSIS

Fig. 2), and in every one of them we seek the problem solution, thus having a “piecewise” approximation to the governing equations. Such discre- tization of the domain requires a different approach for solving the differential equations. Since we are interested now in an area rather than a point, the differential equations must be recast in an approximate integral form which is the basis for the FEM formulation. Several methods are available for reducing the differential conservation equations to suitable integral equations. These include (Huebner & Thornton, 1982):

the direct approach, based on the direct stiffness method of structural analysis the variational approach, based on the calculus of variations the weighted residuals approach, based on Ga- lerkin’s method the energy balance approach, based on the balance of thermal andlor mechanical energy of a system

The overwhelming majority of finite element formulations are based on the variational approach, especially in structural mechanics where the method of virtual work has been the most popular (Martin & Carey, 1973). In fluid mechanics the Galerkin method is usually used (Huebner & Thornton, 1982; Taylor & Hood, 1973; Tanner, 1973; Nickell, Tanner, & Caswell, 1974). Both methods result in the same form of discretized equations. The primitive variables are the velocities v, (or u) and vy (or v) and pressure p (u-v-p formulation, Taylor & Hood, 1973).

VELOCITY-PRESSURE FORMULATION

s y FIGURE 2 Twc-dimensional domain divided into triangular ele-

I x

The virtual work method when applied to the continuity and momentum Eqs, (1) and (2) gives the following integral equations in Cartesian index notation (Martin & Carey, 1973; Mitsoulis, 1984):

where oij is the total stress tensor given by Eq. (9, T~~ is the extra stress tensor, Eij is the rate-of-strain tensor, vi is the velocity vector, 6vi is the virtual velocity vector, 6p is the virtual pressure, and A is


the area of the domain. On the portion of the boundary I‘ where velocities are specified the virtual velocities vanish. The term “virtual” applies to arbitrary changes of the field variables.

We have made use of the definition of the rate- of-strain tensor iii

which for two-dimensional problems reduces to the following vector

109


A constitutive equation relates the stress vector

(10)

where [D] is a constitutive matrix and (ro) a stress vector that accounts for all possible stresses present in the fluid that cannot be included in the product [D]{;]. For a generalized Newtonian fluid we have

la) to (8 (01 = [DIG1 + (rol

thus

where 7 is the viscosity. For a viscoelastic fluid (ro) may contain several terms such as extra rates-of- strain, stresses andlor their derivatives, integrals, etc.

Equations (6) and (7) must now be discretized by an appropriate choice of approximation functions for the primitive variables u, v, p. Several options are available (Huebner & Thornton, 1982). Here we consider a six-node triangular element (see Fig. 3). The behavior of the unknown field variables over each element is approximated by continuous functions expressed in terms of the nodal values of the field variables and sometimes the nodal values of their derivatives up to a certain order. The functions defined over each finite element are called “interpolation functions” or “shape functions.” Mathematical considerations require that the interpolation functions for velocity should be higher by one order than the interpolation functions for pressure. Thus, a quadratic variation is chosen for velocities u and v and a linear variation for pressure p at the vertex nodes. Thus, we have

110

where <> denotes a row vector, [I denotes a column vector and N, are the “interpolation (shape) functions.” These are related to the “natural (area) coordinates” Li for a triangle by the expres- sions

N, = Ll(2L, - l), N4 = 4LlL2

N2 = L2(2L2 - l), N, = 4L2L3 (13a)

N3 = L3(2L3 - I), N6 = 4L3Ll

NP = Li . ( 13b)

The natural coordinates Li are defined by

Ai L. = - ’ A ‘ (14)

It follows that

Therefore, only two of the area coordinates L,, L, and L, are independent.

The relation between area coordinates Li and global coordinates x and y at any point P is given

L1+L,+L3=1. (15)

Inverting

(17)

The above choice of interpolation for u and v dic- tates a linear interpolation for the derivatives of velocities. The rate-of-strain vector {i) is then approximated by I

where [B] is the rate-of-strain matrix.

VOL. 4, NO. 2


Substitution of the approximations (12) and (18) back to the integral Eqs. (6) and (7) and also taking into account the constitutive relation (10) gives a final matrix equation of the form

[KlnlXI, = [F),

or

where

In the above, [K] is a symmetric stiffness matrix, (XI is the vector of nodal unknowns and (FI a load vector containing body forces, surface tractions, forces due to extra stresses {701 and the boundary conditions. The matrix integrations that appear in Eqs. (20) are performed numerically. Equation (19) is derived for the nfh element. By assembling the contributions from all the N elements and by imposing the appropriate compatibility of the nodal unknowns we derive the global stiffness matrix and global load vector. The final set of equations can be solved by a standard matrix solver (e.g., Choleski decomposition). For non-New- tonian problems, the nonlinear set of equations must be solved by using some iterative technique (e.g., Newton-Raphson). For a generalized New- tonian fluid, a direct substitution method is recommended (Picard method) rather than the Newton-Raphson method. The latter is superior for fluids with a constant [D] matrix and a non- zero (701-

In Appendix A the reader may find an illustra- tive example of the application of the finite element method for the simple case of Poiseuille flow between two flat plates. The differential equation is cast in an integral form and the derivation of the stiffness matrix and load vector is demonstrated for a simple linear element.


FIGURE 3 A six-node triangular element for u-v-p formulation.

X

TEMPERATURE FORMULATION

Using the virtual work method the differential equation for conservation of energy (3) is written in an integral form as - -7

+ pC,(GT)vi- dA ax, "'1 (21)

A

where T is the temperature, 6T is the virtual temperature, k is the thermal conductivity, p is the density, C, is the heat capacity, vi is the velocity vector, Q is the rate of heat dissipated (Q = T:VT), and q is the heat input per unit area on boundary rq. Note that 6T = 0 on the part of the boundary where T is specified.

The flow field can be discretized into three-node triangular elements using linear interpolation functions for temperature

T = NTT, + NTT, + NTT, = <NT>(T) (22)

where <NT> is a row vector containing the shape functions (linear functions of position) and [TI is a column vector containing the nodal temperatures. To maintain grid and nodal compatibility between the u-v-p and the T formulations and also improve accuracy of the lower order of approximation for temperatures, each six-node velocity-pressure tri-

1 1 1


FIGURE 4 Subdivision of a 6-node u-v-p triangle into four 3-node linear T- triangles.

X

angle can be subdivided into four temperature triangles (see Fig. 4).

The finite element equation for energy is obtained by substituting Eq. (22) into Eq. (21) yielding for the nth element (Huebner & Thornton, 1982)

(23)

where [H,] is the thermal conductivity matrix, [C.] is the heat capacity matrix, and IQn) is the heat load vector. These are given by

([H,] + [C,IJ(T) + IQ,) = (0)

dA (24a)

Qi = -SNiQdA + NiqdI’. (2W A d

(Note that Cij is a nonsymmetric matrix.) Following the usual assemblage procedure for

the contributions from all of the elements we obtain the matrix equation

[KTI~TJ = IFTI (25)

where [KT] is the global heat stiffness matrix (nonsymmetric), IFT) is the global heat load vector containing boundary conditions and viscous dissipation, and (TI is the vector of the unknown nodal temperatures. The linear system of equations can

112

then be solved by a standard matrix solver for nonsymmetric matrices (e.g., Choleski decomposition).

For a complete nonisothermal analysis of flow problems with temperature dependent properties, the two formulations can be used in an intertwined manner to update the flow and temperature fields. Convergence of the overall systems of equations cannot be guaranteed, in general. However, it has been observed that in some well-defined polymer processing problems, satisfactory convergence was obtained after 2-3 iterations between the continuity-momentum (u-v-p) and the energy (T) formulations (changes in T less than 0.5”C, Mitsoulis, 1984).

The major difficulty in the numerical solutions of the energy equation arises from the presence of the convective term. Past experience both with the FDM and the FEM indicate that spatial oscillations of the solution appear when convection becomes important. The influence of convection in heat transfer analysis is characterized by the Peclet number defined by

where V is a characteristic velocity and H is a characteristic length. The Peclet number represents the ratio of heat transferred by convection to the heat transferred by conduction. For polymer melt processing the Peclet number will be high (in the order

Oscillations are exhibited at moderate to high Pe and are due to different mathematical characteristics of the first and second order operators (i.e., viaT/axi and VT) involved in the energy equation.

To suppress the undesirable oscillations, either the grid density must be increased wherever rapid temperature changes are expected or “upwind finite elements” should be used (Mitsoulis, 1984; Stolle, 1982; Christie, Griffiths, Mitchell, & Zienkiewicz, 1976; Heinrich, Huyakorn, Zienkiewicz, & Mitchell, 1977; Hughes, 1979). The method of “upwinding” has also been used in finite differ- ences (Roache, 1976). In the FEM method, “upwinding” consists of numerically integrating Eq. (24b) at specially selected “upwinding integration points” A, B, C instead of the usual “Gauss- Quadrature integration points,” a, b, c as illustrated in Figure 5 . The optimum distance of points A, €3, C depends on the local Peclet number. The method can be seen as a weight factor approach

VOL. 4. NO. 2

of 102-105).


usually employed for stabilizing a solution process. It should be noted that although upwinding suppresses oscillations, it also decreases accuracy. A remedy is, of course, a more refined grid at the expense of computations. A more detailed study of upwinding for triangular elements is given in Mitsoulis, 1984.

STREAM FUNCTION FORMULATION

In fluid mechanics the stream function $ and the vorticity w are of great importance. For two- dimensional flows these are defined by

It can easily be shown that + and w satisfy the Poisson equation

w . (29) v2* = -

The above equation is valid for any two-dimensional flow of any fluid. Its usefulness lies in the fact that enables us to visualize the flow field by constructing streamlines from the stream function *.

The same finite element formulation used for temperatures (linear 3-node triangles) can also be used for the stream function 4, by realizing that the energy equation with no convection satisfies a Poisson equation of the form

kV2T = -Q (30)

where Q = ?:VV is the viscous dissipation term. Comparison with Eq. (29) shows the equivalence for k = 1, T = $, and Q = w . A solution of a flow field (u-v-p formulation) gives w , and the stream function can, therefore, be obtained a posteriori from the FEM with the appropriate boundary conditions.

FREE SURFACE FLOWS

Flows with free surfaces in polymer processing are encountered whenever a fluid enters or exits from a processing device (e.g., calendering) or flows freely in the atmosphere after being pro- cessed in a confined region (exit flow from dies).


3

2

FIGURE 5 Upwinding scheme for triangular elements.

These problems are particularly difficult to solve since not only the flow characteristics (velocities, pressure, temperature, etc.) are unknown but also the location of the free surface that forms part of the boundary is unknown. The only condition that makes this class of problems solvable is the fact that no fluid can flow through a free surface, which is a streamline. Hence, at the free surface

n - v = O (3 1)

where n is the unit normal vector to the surface and the velocity vector.

The FEM is very attractive for free surface problems because of the ability of the elements to take the shape of the freely flowing fluid. The general procedure is to assume at first a location for the free surface (usually an extension of the solid boundary line that the fluid last encountered) and solve the system of conservation equations for these boundaries. Once the velocity field has been found, a streamline can be constructed at the free surface based on the velocities v, and v, found on that surface and the equation that defines a streamline, i.e.

- -

Integration of Eq. (32) can be carried out (usually numerically by applying Simpson’s rule) to locate a new surface h(x) which is given by

V I

h(x) = h, + s L d x 0 vx

(33)

where h, is the initially assumed location of the free surface and I its length in the x-direction.

113


45O-tapered ¶.on0

entry flow of a power-law fluid through a slit die.

8.000

7.000

6.000

5.000

4.000

3.000

E 4 2.000 Y

With the new boundary known, the system of conservation equations is solved again in the new domain. Recent velocities are used to compute a new surface and the process is repeated until no change in the location of the free surface is observed. Three to five iterations are usually suffi- cient for convergence if the initial free surface is judiciously chosen.

The whole process can be seen as a series of solutions for different problems, although we consider only one material and one process. Such calculations are very difficult and time consuming, even with today’s computers, and present many problems because of computer memory requirements and other complexities imposed by the finite element method.

I I I I I I I I I I I

- + + + -D --- - = = = * A - -- - c - - - -__)

-__) -- -- - - - _.)- - - - + -w 4 * A d - -

# - - - * * * * A

- * Z I I #

- * r r r 0

- D r r r l

PS : xXy= 5000 - D m

VISCOELASTIC FLUIDS

Calculations with viscoelastic fluid models are performed iteratively by starting from inelastic constitutive relations (Le., Eq. (10) with kO) = 0). The main difficulty encountered is the lack of convergence when the elastic effects are comparable to the viscous effects. The elasticity level is usually

expressed in terms of Deborah number (De), Weissenberg number (Ws) or the stress ratio (SR), .which is also referred to as recoverable shear. These dimensionless quantities can be shown to be identical under certain conditions. Here, we define

x Deborah number De = -

where A is a characteristic fluid time and 8 a characteristic process time,

(34) e

V H

Weissenberg number Ws = A- (35)

where V is a characteristic velocity and H a characteristic length, and

(36) “W Stress ratio S, = - 27,

where (NJW is the first normal stress difference and 7w the shear stress, both evaluated at the wall.

Finite element calculations with viscoelastic fluid models require very large computer times and most of the results published in the literature are of little practical significance. Recently, Mitsoulis (1984) proposed a short-cut method for viscoelastic fluid flow calculations. This method is applicable to problems with a dominant flow


direction (Le., flows through long dies). Mitsoulis uses constitutive models that are valid in visco- metric flow with experimentally measured shear and elongational viscosities and normal stress co- efficients. Successful simulations include the pre- diction of vortex growth in die entry flows and extrudate swell.

SOME FEM APPLICATIONS IN POLYMER PROCESSING

The preceding analysis has been successfully employed in a FEM computer program called MACVIP (Mitsoulis, Vlachopoulos, & Mirza, 1983b) that was developed for the solution of two- dimensional and axisymmetric heat and flow problems of non-Newtonian inelastic and viscoelastic fluids in polymer processing. Problems that have been solved include the slider bearing problem, driven cavity flows, tapered entry flow in a die, entry flows in sudden contractions, exit flows from slit and capillary dies, extrudate swell, and a full two-dimensional analysis of calendering that includes the determination of the melt bank free surface. Some of the results are illustrated in Figures 6-12.

Figure 6 presents the flow field for a

45"-tapered entry flow of a polystyrene melt through a slit die and Figure 7 the streamline pat- tern for entry flow in a 1O:l sudden contraction. Such analysis can be used for die design. Figure 8 shows the final finite element grid for the exit flow problem from a slit die and the determination of the extrudate swell for a polystyrene melt. Figures 9 and 10 show the velocity vectors and streamlines. Figures 11 and 12 show the streamlines and isotherms in the melt bank for rigid PVC calendering. These figures are shown here merely to illustrate the type of results obtained by our finite element program. Detailed information on the problems examined can be found elsewhere (Mitsoulis, Vlachopoulos, & Mirza, 1984; and two forthcoming articles).

From the foregoing discussion it is apparent that in the application of the FEM, large matrices are handled and very large numbers of arithmetic operations are performed. Consequently, the computer memory and time requirements are large, especially for nonlinear andlor free surface problems. Some typical CPU times are shown in Table I, for McMaster's CDC 64001CYBER 170 (available memory 377 K). It should be pointed out that the solution of the flow field in geometries such as sudden contractions and sudden expansions is particularly difficult because of stress singularities (mathematical discontinuities) at the entrance cor-

~

FIGURE 7 Streamline pat- tern for entry

fiowina10:l planar sudden 1.0 I I I I I I I I I contraction of

POLYSTYR E N E : De= 5.47 (t:507~-')

.8

- 6

.4

.2

.O

-.Z

-.I

- a 6

-.8

a polystyrene melt.

-1.0 I I I l l I 1 I I I I 1 -2.0 -1.6 -1.2 -.8 - . I .O .4 .8 1.2 1.6 2 .o

X-COORDINRTE* 1 CM I

ADVANCES IN POLYMER TECHNOLOGY 115

FINITE ELEMENT METHOD FOR FLOW AND HEAT TRANSFER ANAL YSIS

FIGURE 8 Final finite element grid for the exit flow from a slit die and the determination of extrudate swell for a polystyrene melt.

3 .O

2 .O

1.0

I

0.0

-1.0

-2 .O

-3 .O I I I I 1 I I I I I -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

OISTFlNCE I X/H

FIGURE 9 Velocity vectors for a polystyrene melt exiting from a slit die.

I :

3 00

2 .O

1.0

0.0

-1.0

-2 a 0

-3 m O

I I I I I I '162.7 PS De=2.7 -1

SLIT DIE

-3 .O -2 .O -1.0 0 00 1.0 2 00 3 .O 4 -0 5 .O

DISTFINCEP X/H

116 VOL. 4, NO. 2


2 .o

1.0

3 .O I I I I I 1 I I I

- PS De=2.7

l+J:0 -2 .o

-3.0 I 1 1 I I I 1 I I 1

-2.0 -1.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

DISTFlNCE I X/ H

FIGURE 10 Streamline pat- tern for a polystyrene melt exiting from a slit die.

1.0

.I

.C

.4

x .z

- .O c

P -.2

0 8 - . 4

u L -.c

-.I

-1.0

-1.2

ROLL 3

Rig id PVC 4

FIGURE 11 Streamline pat- tern in the melt bank for PVC calendering (from Mit- soulis, Vlacho- poulos, Mirza, “Calendering Analysis”).

FIGURE 12 Isotherms in

.I the melt bank for PVC calendering (from

Vlachopoulos, Mirza, “Calen-

ysis”).

1.0

.c

I Mitsoulis,

I .z

c - .o dering Anal-

P -.I .. u .g ’.I

L - . c

-.I

-1.0

-1.2 -5.7 -5 .5-5.3-5.1-~.Y-~.7-~.5-~.1-~.1-1.Y-1.7-~.5-1.1-1.1-2.¶-2.7-2.1-2.1-2.1-1.Y-1.7-1.5-1.~-1.1 -.¶ -.7 -.5 -.I -.I .I .I .5

X-COOROIWTEI CM



ner to the die and at the die exit. A large number of elements must be concentrated in these areas to better capture the irregular behavior. The inclu- sion of viscoelasticity through a standard constitutive equation requires numerical differentiation of velocities and velocity gradients which may introduce additional errors. To avoid this, some re- searchers, notably M. J. Crochet of Louvain la Neuve, Belgium, and R. A. Brown and R. C. Armstrong of MIT, use the stresses as primary variables along with velocities and pressure. This amounts to six unknowns per node for the two- dimensional (u, v, p, 7,,, 7yy, 7J and seven unknowns per node for the axisymmetric problem (u, w p, T,,, T ~ , T,, 7& The total number of nodal variables can therefore become very large and sometimes prohibitive for engineering calculations. For example, Crochet’s results for the extrudate swell of a viscoelastic model fluid called Oldroyd-B were obtained with a grid consisting of 75 quadrilateral elements, 357 nodes, and a total number of 1889 variables for a slit die and 2246 variables for a capillary die. The total running time was reported to be 50 CPU hours on a PRIME 750 computer. Brown and Armstrong have used up to 7000 variables with commensurate time and memory requirements. Standard solvers

for banded nonsymmetric systems are inappro- priate for such large numbers of nodal variables. The frontal method of solution (Irons, 1970; Hood, 1976) with diagonal pivoting is then applied which leads to a drastic reduction of core storage. Nevertheless, while memory requirements can thus be overcome, computer run time remains very high.

CONCLUDING REMARKS

While there are still many problems to be ironed out, the finite element method is a very powerful tool for solving flow and heat transfer problems. It appears to be the only rational method of solution for problems involving complicated geometrical boundaries and complex interactions of flow and temperature fields. The development of user-friendly computer packages will make the method accessible to persons that have a limited background in variational calculus or matrix algebra. The wide use of such a powerful tool will enhance our understanding of polymer flow, mixing and melting and may spear- head new dramatic developments in polymer processing.

TABLE I Typical Requirements for Problems Solved by the FEM (MACVIP Program) Using a CDC 64OO/CYBER 170

Problem Fluid Free Number Number Number Computer Run Overall Surface of of of Memory Time per Run

Elements Nodes Unknowns Required Iteration Time (k) (5) (hr)

Tapered Entry Flow Inelastic Power-Law No 64 153 317 260 70 0.15 Isothermal

Sudden Entry Flow Inelastic Power-Law No 200 459 881 350 200 0.4 Isothermal

Exit flow- Extrudate Swell

Viscoelastic Isothermal

Viscoelastic Isothermal

Calendering of Rigid PVC Inelastic

No 200

Yes 184

118 Power-Law Non-isothermal

Yes

Slip at wall

459

423

28 1

88 1

858

588

350

360

345

200 0.3.

225 0.5*

120 2.5

*For each elasticity level (Le., for each Deborah number).

118 VOL. 4, NO. 2

FINITE ELEMENT METHOD FOR FLOW AND HEA T TRANSFER ANALYSIS

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Irons, D. M. 1970. A Frontal Solution Program for Finite Element Analysis, Int. J. Numer. Methods Eng. 2: 5 .

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Mitsoulis, E. 1984. Finite Element Analysis of Two-Dimensional Poly- merMelt Flows, Ph.D. Thesis, Dept. Chem. Eng., McMaster Univ., Hamilton. Ontario. Canada.

Mitsoulis, E.. Vlachopoulos, J., and Mirza, F. A. 1983a. Finite Element Analysis of Two-Dimensional Polymer Melt Flows, Po/ym. Proc. Eng. I : 281.

Mitsoulis, E., Vlachopoulos, J., and Mirza, F. A. 1983b. MACVIP- A Finite Element Program for Creeping Viscoelatic Flows, Internal Report. Faculty of Engineering, McMaster Univ., Hamilton, Ontario, Canada.

Mitsoulis, E., Vlachopoulos, J., and Mirza, F. A. 1984. Numerical Simulation of Entry and Exit Flows in Slit Dies, to appear in Polym. Eng. Sci. 24.

Mitsoulis, E., Vlachopoulos. J., and Mirza, F. A. A Numerical Study of the Effect of Normal Stresses and Elongational Viscosity on Entry Vortex Growth and Extrudate Swell, to appear in Polym. Eng. Sci.

Mitsoulis. E., Vlachopoulos, J., and Mirza, F. A. Calendering Analysis without the Lubrication Approximation, to appear in Polym. Eng. Sci.

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APPENDIX A

Poiseuille Flow by FEM

In this Appendix we illustrate the application of the FEM for the case of Poiseuille flow of a New- tonian fluid between two flat plates. Our objective is to demonstrate how a variational principle is used and how the stiffness matrix and the load vector are generated.

For the analysis we consider a simple linear element, shown in Figure A1 and designate ends 1 and 2 as “nodes.”

The governing differential equation is

dp- d2u dx ’dyz -_

where dpldx = p’ is assumed uniform across the element length L. The nodal values of u(y) are

u(0) = u, (A21

u(L) = u2 (A31

and are called “nodal unknowns” or “degrees of freedom.” We want to find the function of u(y) that satisfies Eq. (Al).


From variational calculus (see for example Schechter, 1967) it is well known that this problem is equivalent to finding the function u(y) that min- imizes the functional

We shall ignore the fact that this problem has an exact solution and proceed to find an “approximate” solution. We approximate the velocity u(y) by the following quadratic polynomial

FIGURE A1 Single linear element for flow between

P

LL 0 .- 1

119


u = a + by + cy2. (AS)

Since we have three unknowns-a, b, c-and two degrees of freedom-u, and u2-we also need to introduce another equation. This is done by differentiating Eq. (AS)

We find now a , b and c in terms of the nodal unknowns

(A74

(A7b)

U I - j.1L)/L2 ( A ~ c )

where we have introduced an extra degree of freedom f l , the unknown velocity gradient. Equations (AS) and (A6) are then written in terms of the nodal unknowns as

u = uI + j.y + 1 /Lz(uZ - uI - j.,L)y2 (A8)

+ 21LZ(u2 - uI - j.,L)y (A9)

Noting that in Eq. (A4) one of the terms is associ- ated with viscosity while the other with the pressure gradient we may write

J = Q + W (A101

+ = du/dy =

where

and

To minimize the functional we must have

or

where xi stands for u,, u2 and Substitution of Eq. (AS) in (A1 1) yields

Q = ~ P ~ [ ~ ~ + ~ ( U ~ - - ~ ~ l L 2 - j . lL )~ ]zd~. (A151

After the appropriate mathematical manipulations we obtain

120

4 4 4 Q = -p ~ j . ; + -u; + -u: + -L+; +

2 [ 3L 3L 3

+ 2f1u2 - 2f,u, - 2 3 L - -j-UIU2 8 -

8 8 - -j-r1u2 + +,ul]

Differentiation yields

---,[2Lf1 a Q - 1 ++ 8L + ar, 2

+ 2u, - 2u, - 4Lf, -

8

Equation (A17) can be written in a matrix form as

or in “shorthand”notation

(Fi) = [KIIxi) -

The matrix [K] is a symmetric matrix and is usually called by historical default the “stiffness” matrix, due to its original derivation in structural mechanics. The vector (xi) is the vector of unknown nodal variables for which we seek the solution.

Similarly, we can substitute Eq. (A8) into Eq. (A12) to obtain


After the appropriate mathematical manipulations we obtain

1 W = p ‘ L

By differentiating we have

-Fi

or

2 -p’L 3

1 -p’ L 3

1 -p‘L2 6

2 =-plLi\ The vector IFi) is then called “load” vector, again from its original structural mechanics derivation.

Thus we have

(stiffness matrix) (load vector)

or in “shorthand” notation

[KlIxiJ = (FJ. (A24b)

Equation (A24) is the standard finite element form that all formulations lead to (i.e.y a matrix equation that includes a stiffness matrix and a load vector).

For this simple case we can proceed and solve explicitly Eq. (A24). A further simplification is obtained if we consider the kinematic boundary condition at the wall, i.e.,

uz = 0 (A251

This eliminates the second row and column in Eq. (A24) which now becomes

ZE” 1 [’: - and finally

p’Lz L2 dp ’ 2p 2p dx

(A27a) u =--=---

i , = o (A27b)

The above derivation has been obtained by using a single linear element with two nodes 1 and 2. It is obvious that more elements can be put together in various ways. In all cases a stiffness matrix and a load vector must be derived for each element. A standard procedure for assembling all the elements together can then be applied that takes care of nodal compatibility. The kinematic boundary conditions and/ or external forces must also be employed. The final assemblage leads to a “global stiffness” matrix and a “global load” vector that can be solved by a standard matrix solver.

It is interesting to note that for this simple problem the exact solution of the differential equation (Al l is

and for the midplane of the channel of width 2L we have

L2 dp I 2p dx

u =--- (A29a)

f, = o (A29b)

(i.e., the two solutions are identical).


the finite element method for flow and heat transfer analysis

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finite element method

finite element equations

flow problems

term finite element

finite elements

field problems

use of fem

irrotational flow