Viscous Dissipation in Shear Flows of Molten Polymers

Viscous Dissipation in Shear Flows of Molten Polymers HORST HENNING WINTER Institut f u r Kunststoff(echno1ogie. Universitat Stuttgarr, Stuttgart, West Germany

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. System of Equations. . . . . . . . . . . . . . . . . . . . . . . . . B. Thermal Properties . . . . . . . . . . . . . . . . . . . . . . . . . C. Rheological Constitutive Equation. . . . . . . . . . . . . . . . . . . 11. Shear Flow (Viscometric Flow). . . . . . . . . . . . . . . . . . . . . . A. Thermal Boundary Condition. . . . . . . . . . . . . . . . . . . . . B. Steady Shear Flow with Open Stream Lines. . . . . . . . . . . . . . . C. Shear Flow with Closed Stream Lines . . . . . . . . . . . . . . . . . 111. Elongational Flow; Shear Flow and Elongational Flow Superimposed (Nonviscometric Flow) . . . . . . . . . . . . . . . . . . . . . . . . . IV. Summary, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

205 207 209 211 212 222 221 250 260 262 263 264

I. Introduction Polymer processing and applied polymer rheology occur at relatively high temperatures and often at high temperature gradients. In molten polymers, large stresses are required to maintain the flow and, additionally to convective and conductive heat transfer, temperatures essentially depend on viscous dissipation, i.e., on conversion of mechanical energy into heat. Velocity and temperature fields influence each other: the temperatures influence the flow through the temperature-dependent rheological properties, and the velocities influence the temperatures through convection, through dissipation, and through anisotropical effects (which are investigated very little) on the thermal properties. 205

HORSTH. WINTER

206

Research in rheology and in thermodynamics related to heat transfer problems is mostly done separately, by rheologists on molten polymers or polymer solutions at constant temperature, and by thermodynamicists on polymers at rest. The difficult task of combining the two areas is left to the polymer engineers (see for instance [1-41). A number of assumptions have to be introduced into the heat transfer analysis before applied problems can be solved. The different flow problems involving heat transfer and viscous dissipation can be classified in groups as shown in Table I. Each of the groups is characterized by different rheological phenomena, and one has to choose very different rheological constitutive equations to describe them. The two main groups are channeljow (includingflow geometries with partly solid and partly free boundary) and free surface flow (with no solid boundary). In polymer processing, channel flow of molten polymers occurs in a large variety of flow geometries. The polymer is forced through a channel by a pressure gradient (flow in an extruder die, for instance),or it is dragged along by a moving wall (rotating screw in a stationary cylinder, for instance). Very often both types of flow are superimposed on each other. Free surface flow for example occurs in film blowing or fiber spinning.

TABLE I

CLASSIFICATION OF

HEATTRANSFER AND VISCOUS DISSIPATION IN MOLTENPOLYMERS heat transfer and viscous dissipation in molten polymers

h free surface flow

channel flow

shear flow shear flow with open streamlines

shear flow with closed streamlines

(shear free flow)

shear flow and elongational flow superimposed (non-viscometricflow)

For rheological reasons, channel flow problems are subdivided into shear (also called viscometric flow) and nonuiscometric flow. The separation of the shear flow problems into one group with open stream lines and one with closed stream lines has to be made since their thermal development is different. Throughout the first section, the heat transfer problem will be considered in general, i.e., the relevant equations are listed in a general form and the properties are described. The rheological properties of the molten polymers have to be formulated differently according to the various flow types since

jow

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

207

the length of the following sections is supposed to reflect the degree of understanding of the respective flow and heat transfer problems. In Section 11, heat transfer in shear flow will be analyzed. A large emphasis will be laid on replacing the commonly used idealized boundary conditions, i.e., constant wall temperature or constant wall heat flux (with the limiting case of the adiabatic wall), by more general conditions. In practical applications, the idealized conditions will rarely occur; actually it is difficult to achieve them even in especially designed model experiments. To make the analysis applicable, heat transfer in a flowing polymer should not be studied separately inside the fluid, but together with the surrounding wall. In this analysis the heat transfer at the wall is described by an outer temperature difference (temperature of the surroundings minus temperature at the boundary) and the Biot number, which otherwise has been used successfully for describing the boundary conditions for temperature calculations in solids. The Biot number is appropriate for describing boundary conditions between isothermal and adiabatical, as they occur in real processes. Additionally, the thermal capacity of the walls is included in the analysis by introducing the capacitance parameter C . Heat transfer in viscometric flow has been studied quite extensively in the literature, and at the present state it seems to be necessary to show the many common aspects of the different studies. Thus, as the main goal of this study a unifying concept will be developed. This concept makes it possible to comprise the most important shearjow cases into a single one, which can be solved with one numerical program. For Section I11 on nonviscometric flow in channels and flow with free boundaries, the description will not go much further than stating the problem, showing the present methods of solution, and listing references. Since nearly all of the results in this report are on shear flow, the title is taken to be “in shear flows” even if the problem is stated in a general form and Section I11 is on nonviscometric flows. Heat transfer in non-Newtonian fluids at negligible viscous dissipation is not included in this report (see instead [5-7]), although it can be treated as a limiting case of the corresponding flow with viscous dissipation. A. SYSTEM OF EQUATIONS

The problems are governed by the equations describing the conservation of mass appt v * (pv) = 0 (1.1)

+

and the conservation of energy p DelDt = V ( k V T ) 9

+ o :Vv,

( 1.2)

HORSTH . WINTER

208

by the stress equation of motion p DvlDt = V * u

+ pg,

(1.3)

and by the constitutive equation which will be described below, together with the appropriate flow geometries. The three equations above are derived and tabulated in textbooks (see for instance [S]) for different coordinate systems. a/a denotes the partial and DID the substantial derivative; V is the “nabla” operator. Density p and thermal conductivity k are properties of the fluid. Velocity v, internal energy e, temperature T , time t, and stress CT are the variables. The stress CT is defined in such a way that the force on the positive side of a surface element of unit area and normal vector n is n 0. The equation of energy says that the rate of gain of internal energy per unit volume ( p De/Dt) is equal to the rate of internal energy input by conduction per unit volume V (k V T ) plus the rate of work by the stress on the volume element u :Vv, which is being partly stored and partly dissipated during the flow. For heat transfer studies, the internal energy has to be defined in terms of the fluid temperature and the strain and stress variables. IncornpressibleJiuid: In rheology the fluid is usually supposed to be incompressible (even when properties such as the viscosity are allowed to depend on pressure). The flow geometry, the temperature, and the rheological properties of the fluid determine the stress completely, except for an arbitrary added isotropic pressure [9]. Therefore, the stress is commonly separated into an arbitrary pressure p and the extra stress 7 , which is defined in the rheological constitutive equation, viz. a = -pd+z. (1.4) 6 denotes the unit tensor. In some flow problems, it is convenient to define the isotropic pressure p to be equal to one of the normal stress components in a certain coordinate system ( p = -ell, p = -crz2, or p = - L T ~ ~ )while , in other flow problems it might be preferable to define p = -(trace a)/3. For an “incompressible”fluid, the change in internal energy and the work of the stress per unit time are determined by p DelDt = cp DTIDt,

u:Vv = - p

v

*

v

+ 7:vv,

(1.5)

and Eq. (1.2) becomes cp DTIDt = V * ( k V T )

+ z:Vv.

(14

The specific heat capacity c is defined as the thermal energy needed per unit mass and Kelvin degree for changing the temperature of a material. Since the density is taken to be constant, c has to be measured at constant

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

209

density. If the fluid were really incompressible, the specific heat should be the same for measurements at constant density (c,) or at constant pressure (c,). From thermodynamic data at rest (Eq. (l.ll)),however, one finds that c, and c, of polymer melts differ by about 10%. Compressible fluid: There are difficulties in relating strain and stress in deforming materials that are slightly compressible. One commonly assumes that the deformation can be separated into two parts: a deformation at constant density and the volume change [lo]. Neglecting the influence on each other, the deformation at constant density is described by the constitutive equation, and the density of the flowing polymer is determined from equilibrium data p(T, p ) measured on the fluid at rest (taking p = -(trace 4/31. There also seem to be difficulties in defining the internal energy e of a compressible flowing fluid: one assumes that e can be described in terms of p and p only, independently of the other stress and strain variables (see for instance [81): e = e(p, P). Applying this relation to the flowing polymer melt, the substantial derivative of the internal energy then becomes De

PK= - P V

*V

+ € TDP -+ Dt

DT pepDt

(1.7)

with E = -p-l(i?p/aT),, the coefficient of thermal expansion. E and c, are evaluated from rest data at temperature T and the “pressure” p = -(trace 4 3 . The equation of energy takes the form usually shown in the literature [ll]: DT DP PC - = V .(k V T ) + ET- + t:Vv. Dt Dt From this equation, calculated temperature fields in channel flow (see Fig. 13, p. 244) show large temperature decreases due to cooling by expansion. The assumptions made above (concerning the density changes and the internal energy) are rather severe, and further experimental studies are needed to investigate their validity.

B. THERMAL PROPERTIES The properties in the analysis are the density p, the specific heat c, and the thermal conductivity k : the thermal diffusivity is defined as a = k/pc. Rheological properties are defined separately in the constitutive equation. In a stationary fluid, the density p ( p , T ) is a function of pressure and temperature. It can be described by the equation of Spencer and Gilmore

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210

TABLE I1 CONSTANn OF EQ. (1.9) MEASURED BY MATERIAL SPENCER AND GILMORE [121

polymer Id.PE

PS

PMMA CAB

b*

m3/kg]

W

p* [N/m']

0.875 0.822 0.734 0.688

3.275 1.863 2.157 2.844

[12]: ( l / P - b*)(P

x

10'

x 10' x 10' x 10'

+ P*)

= RT/W,

kg/g-mole] 28 104 100

54

(1.9)

where b*, p*, and W are material constants. Their values are tabulated (Table 11) for some examples of the most widely used polymers; R = 8.314 [J/K g-mole] is the gas law constant. From Eq. (1.9) one can evaluate the term ET of the equation of energy ET = 1

since b* is always smaller than

p-l,

- pb*;

(1.10)

the dimensionless product CT adopts

positive values smaller than unity.

The density generally is measured on the polymer at rest and in thermodynamic equilibrium. Dynamic measurements by Matsuoka and Maxwell [131, however, show a very delayed response of polyolefines to sudden pressure changes. Thus, the use of equilibrium density data restricts the analysis to flows of slowly changing pressures. The reaction to temperature changes is similarly delayed [141. Additionally the flow might influence the density. The specific heat commonly is measured at constant pressure. Using the equation proposed by Spencer and Gilmore, Eq. (1.9),one can determine the c, from specific heat data at constant pressure specificheat at constant density.~ C, =

cP - R/W.

(1.11)

The thermal conductivity k and the specific heat capacity cp are slowly varying functions with temperature and they also depend on pressure. In flowing polymers the thermal conductivity possibly varies with direction. For most polymers the temperature dependence can be expressed in a linear form

k

-

=

E(1 + uJT - To)),

cP = Tp(l + a,( T - To)).

(1.12)

k and Fp are values at some reference condition (temperature T o ) ,while ak and a, are the temperature coefficients, which might be positive or negative

VISCOUS DISSIPATION IN FLOWING MOLTEN POLYMERS

21 1

TABLE 111

THERMAL CONDUCTIVITY AND SPECIFIC HEAT CAPACITY OF SOMEMOLTEN POLYMERS AT TEMPERATURES To (see [ 15 -231) To

(‘P

k

Polymer

rC]

[lo3 Nm/kg K]

[N/K s]

1.d.PE h.d.PE

150

2.51 2.65 2.80

0.241 0.255

PP PVC

150 180

100

1.53

150 150

2.04

PMMA

0.166 0.167 0.195

PS

-

-

depending on the polymer in question and on the temperature range. Table I11 shows some values of k and c p .More detailed data on the properties can be found in references [15-231, for instance. C. RHEOLOGICAL CONSTITUTIVE EQUATION For a large number of fluids, which can be regarded as incompressible, the stress can be described by the Stokes equation u = -p6

+ qy,

(1.13)

the simplest tensor generalization of Newton’s law of viscosity. Here p is the isotropic pressure, and i, = (Vv) + (Vv)’ is the rate of strain tensor; the viscosity q depends on temperature and on pressure, but not on the time t or on any kinematic quantities such as 9. Fluids that show this behavior are called Newtonian. The Stokes equation has been generalized by taking the viscosity q($) to be a function of the second invariant of the rate of strain tensor [24] : u = -p 6

+ q($)i,;

i,

= (+i,:i,)l’Z.

(1.14)

This equation defines the “generalized Newtonian fluid.” It has been applied quite successfully to molten polymers in steady shearjlow for calculating the shear component of the stress tensor in an appropriate coordinate system; however, the normal stress components calculated from this equation are known to be unrealistic for molten polymers. In general, the equation might be misleading in its tensor form because it does not allow one to calculate meaningful stress components in arbitrary coordinate systems. The appropriate statement of the rheological equation for molten polymers in steady shear flow is given by Criminale et al. [25]; their equation will be applied in Section 11.

21 2

HORSTH.WINTER

The rheological properties of elastic fluids (such as molten polymers) at any given position depend on the strain and temperature history of the fluid elements when they arrive at that position, independently of the history of neighboring particles. Translational and rotational movements do not influence the stress [9]. Depending on the type of flow, the rheological behavior of molten polymers is more or less different from the behavior of Newtonian fluids. Up to now there exists no general constitutive equation to describe all the phenomena known for a given polymer melt. Additionally, the temperature effects on the rheological properties have not been studied at all or only at different levels of homogeneous temperatures. The constitutive equations used will be stated in the beginning of Sections I1 and 111. 11. Shear Flow (Viscometric Flow)

In shear flow at constant density, material surfaces move “rigidly” (i.e., without stretching) across each other. These surfaces are called shear suduces [26]. Pipe flow, which is an example of shear flow, sometimes is called telescopic flow since its rigidly moving surfaces are concentric cylinders. In Fig. 1 the deformation of particle Po at the origin is described by the relative motion ofneighboring planes. On the left (Fig. la) an orthonormal coordinate system is chosen, such that x1 = direction of shear, x2 = direction of velocity gradient,

xg = neutral direction.

shear direction

tan

D

E

x21

= dtan

,shear direction

E,J*+

[tan c3d2 ‘

FIG.1 . Unidirectional shear flow in Cartesian coordinates. Shear flow in the 1-3 plane as superposition of shear flow along x, and along xj.

VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS

213

The direction ofshear (or shear direction) and related quantities are defined following [26]: an orthonormal coordinate system is chosen to have its x1 axis and its x3 axis in a shear surface (see Fig. 1). The x2 axis is perpendicular to the shear surface; it projects point Po onto neighboring shear surfaces. During shear flow, shear surfaces move past each other, and the normal projection of Po draws a line on neighboring shear surfaces. This line is called the shear line. (Note: In the examples of Fig. 1, the shear surfaces are planes, and the shear lines are straight.) The tangent to the shear line at time t is defined to be the shear direction at time t, and the angle of the shear line with the x1 coordinate is the shear angle cp; if the direction of shear remains constant with time, the flow is called unidirectional. For most applications, the direction of shear is identical with the direction of flow; an exception is the othogonal rheometer [26, p. 761, for instance. The shear rate, the extra stress, and the temperature are supposed to be uniform in the direction of shear, but they may change perpendicularly to the direction of shear (even within shear surfaces). If one allows for shear in the 3 direction (Fig. lb) additionally to the shear in the 1 direction, the shear direction is in the 1-3 plane (Fig. lc); for some steady shear flow geometries (as in helical flow), it is convenient to choose a global coordinate system with the velocity vector in the 1-3 plane and the velocity gradient normal to the 1-3 plane. The matrix of the rate of strain tensor becomes

[tl

=

[

Pl2

2:

:23]?

(2.1)

Q23

and the second invariant of the rate of strain tensor defines the shear rate: In unidirectional shear flow at constant temperature T o ,constant pressure p,,, and constant volume, the stress is given by the shear rate Q(t) and the three shear-rate-dependent viscometric functions [25,26] : viscosity q ( j t T o ,po), first normal stress coefficient m , T o ,po), second normal stress coefficient $2(QCm, T o ,po). Here symbolizes the “history of shear rates” to which the medium was submitted up to the present time t. Up to now, there do not seem to exist heat transfer studies on shear flow in general; however for steady shear flow, the publications are numerous. In Section II.C.3 an example of heat transfer in unsteady unidirectional shear flow will be shown.

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214

Steady Shear Flow and Shear Viscosity

For unsteady shear flow the shear direction and/or the value of the shear rate change with time; the shear surfaces, however, are maintained. If the shear direction and the shear rate are kept constant over some time (cp = const; j~ = const), the shear stress approaches a constant value, i.e., the viscosity adopts a constant value:

?(t,To,P o ) = lim ?(?‘-m, t+m

To,P o ) ,

(2.3)

called the viscosity, the shear viscosity, or the apparent viscosity. The flow becomes steady (unidirectional)shearpow. The index on To and po indicates that the viscosity is dejned at constant temperature and pressure. The elastic properties of the fluid are represented in steady shear flow by the two viscometric functions t,bl, t,b2. These functions, however, do not have any influence on heat transfer and viscous dissipation; this will be shown in the following. The only two terms in the system ofequations (Eqs.(1.1)-(1.3)) that contain the stress are a:Vv and V u. The rate of work by the stress a:Vv, which in steady shear pow is dissipated completely, sometimes is called the dissipation function. Taking the coordinate system of the shear flow, one can evaluate the two terms [8, p. 7381; u:Vv is a scalar 0:Vv = 212?12 + 2 2 3 7 2 3 , and V u is a vector with the three components

(2.4)

-

[V.U]l

=

[V-a], =

[V

.I3

=

The stress u has been decomposed into p and z as shown in Eq. (1.4).If x1 is the shear direction (Fig. la), the 1component of V u is used for calculating the velocity and the pressure gradient; due to the symmetry of the flow (at/ax, = 0); and since 7 1 3 = 0, the 1 component reduces to

-

[v*u]l =

--a P ax,

+ -.ax,

a212

For a shear direction in the 1-3 plane (Fig. lc),the 1 and the 3 components are used for calculating the velocity and the pressure gradient; since az/ax, = 0

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

215

and &/ax3 = 0, they reduce to [V.UIl

=

--aaPx ,

+ -,a x ,

az12

[V*6I3 =

--

ax3

+ax,

(2.7)

For calculating the velocity, the pressure gradient, and the rate of work by the stress, one finds from Eqs. (2.4), (2.6), (2.7) that one needs only the shear components z12,zZ3of the stress matrix 212

= q(?,

912,

723

= q(?,

(2.8)

j23*

The shear component 213 = sin 2q1($~+ $,)p2/2 and the normal stress components do not contribute to the analysis. The viscosity q is the only rheological property needed for solving heat transfer problems in unidirectional steady shear flow. Thus the heat transfer analysis of Section I1 is not restricted to purely uiscousftuids, even if the normal stresses are not mentioned further. The pressure and the normal stresses can separately be determined from the pressure gradient, the 2 component of the stress equation of motion (which contains [V * el2),and the appropriate boundary conditions. Some typical curves of viscosities referring to different temperatures are shown in Fig. 2 [27]. At low shear rates (p < 10 s-’) one measures the viscosity in Couette or in cone and plate rheometers, and at high shear rates ( j ~> 1 s-’) one uses a capillary or slit viscometer; the temperature in the test section is kept as uniform as possible.

shear

rate 7

10’

Is-’]

10’

103

l@

FIG.2. Typical viscosity curve of molten polymer (low density polyethylene) measured by Meissner [27].

HORSTH. WINTER

216

The pressure and temperature dependence of the viscosity usually is described by the corresponding coefficients pressure coefficient temperature coefficient

=

-1

(3) .

v aT

(2.10)

b,P

(Note: Some authors define the temperature coefficient p at constant shear stress instead of at constant shear rate.) In the expression for the viscosity, the variables are separated. The pressure dependence is described by an exponential function. For incorporating the temperature dependence one may take just an exponential function

v(P, T, PI

= f ( P 9 P ) exp( - BT)

(2.1 1)

or use an Arrhenius type expression V(P, T, P) = f(P, P ) e x p ( E / W

(2.12)

whose “activation energy” E has been reported to be a material constant over wide temperature ranges [28, 291. The temperature coefficient of the viscosity at temperatures around To can then be determined as (if one expands (EIRT) around To) P(T0) = E/RTo2

(2.13)

i.e., for constant E, the temperature coefficient is proportional T-’. The temperature dependence of P normally is neglected in analytical studies, which is acceptable if the deviations AT from the temperature level T o are not too large. The relative difference of the two expressions, Eqs. (2.11) and (2.12),is

= 1

- exp[-fl(AT)2/(To

+ AT)];

(2.14)

as an example AT = 10 K, fl = K-’, T o = 400 K gives a relative difference of less than 0.25%. At very low shear rates (p < 10-1-10-3 depending on temperature, polymer, molecular weight distribution) the viscosity adopts a value inde-

VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS

217

pendent of shear rate, the zero viscosity qo(T,p ) (see Fig. 2). At medium and high shear rates (jJ > lo), as they occur in polymer processing, the viscosity curve q ( j ) is nearly a straight line in the log-log plot. Ranges of the curve can be approximated by a power law [30] which will be formulated as

(2.15) The power law exponent is different for different ranges of i), T , p ; for molten polymers, the value of m is between 2 and 5. In Eq. (2.15),q = q(T, p o , T o ) is a reference viscosity within the power law region, i.e., q at the reference shear rate 7, at the reference pressure p o , and at the reference temperature To.In the literature the power law model has been used very widely because its form allows direct integration of the equation of motion for several flow geometries to be carried out. There are very few data on CI and p available; however, a0 and Po values of the zero viscosity ro have been published for several molten polymers. Thus, a relation will be derived in the following between CI and a, and between p and Po. The viscosity curves measured at different levels of T and p can be condensed to a single one, the so-called muster curue [31,32] q(B3

7-9

p)/qo(T,P) = f (jJ

*

?O(T

P) 1.

(2.16)

In the transformation process, the viscosity curves are moved in the log-log plot in the direction of -45“; the shape of each curve remains the same. Therefore, the shape of the master curve is identical with the shape of the other viscosity curves. At larger values qoi) (> lo4 N m-’), ranges of the master curve can be fitted again by the power law shown above q ( k T, p)lqo(T, P) = w J q o ) ( l / m ) -l ,

(2.17)

and the viscosity becomes

~ ( 9 T, , p ) = f+h/mj(l/m)-l.

(2.18)

K is a “material constant” whose dimension depends on the value of the power law exponent m ; K will be replaced by introducing the reference viscosity Fj of Eq. (2.15). For certain ranges, the power law exponent m is independent of temperature and pressure (since the master curve is independent of temperature and pressure), but it slowly increases with higher values of jq0 ranges. The pressure and temperature dependence of the viscosity q is comprised by the pressure- and temperature-dependent zero viscosity qo only. The pressure coefficient a,the temperature coefficient B, and the activation energy E of the viscosity in the power law region are then related to

HORSTH. WINTER

21 8

TABLE IV E , AND PRESSURE COEFFICIENT a,, OF ACTIVATION ENERGY THE ZERO VISCOSITY qoo

Polymer

a0

EO

[lo4 J/gm mole]

1.d.PE

5.44

h.d.PE PS PMMA

7.08-8.99 -

[lo-’ m2 N-I] -

3.25-3.98 4.28-9.07 2.45-4.08

14.23-19.1 3

Literature 1281 C331 ~321 1331

The temperature coefficient jo of the zero viscosity around the temperature To is equal to E o / R T o 2 ,see Eq. (2.13). a and j of the power law region can be determined from a. and Po by dividing with m, see Eq. (2.19).

ao, Po, and Eo of the zero viscosity by a = ao/m,

P

=

Po/m,

E = Eo/m.

(2.19)

Table IV [28,32,33] lists E o and a. values of some polymers; the data can be used to determine a and P of the power law which describes the viscosity in the jqo range of the application in question. Semjonow [29] collected Eo data from the literature which is quite extensive. Due to the small values of a, the pressure dependence of the viscosity usually can be neglected up to moderate pressure levels ( < 300 bar, depending on the value of a) as they occur in extrusion. In injection molding studies, however, the pressure may adopt values up to 1500 bar and the pressure dependence should be included. In the following study, the pressure dependence of viscosity is not taken into consideration, although the numerical procedure would not have to be much different: it just would require one or more iterations of the whole computation procedure, until the axial pressure profile along the channel is known. The concept of steady shear flow is a theoretical one and it can only be approximated. However, the rheological properties of molten polymers do not seem to be too sensitive to some deviation from steady shear flow; and for a large number of applications, the results from steady shear flow calculations agree with flow experiments reasonably well. The shear flow might be unsteady (a/& # 0 ) ; during startups a constant stress is achieved only after some time of development. But even if the flow is steady (a/& = 0), it still might deviate from shear. Deviations occur as slowly relaxing entrance effects; temperature changes along stream lines, which induce changes in shear rate, are in contradiction to “steady shear flow,” which is defined to be isothermal and at constant shear rate; in Poiseuille flow, pressure changes

VISCOUS

DISSIPATION IN FLOWING MOLTENPOLYMERS

219

actually influence the density, while the fluid supposedly is incompressible. Flows of this kind are called nearly steady shearjlows, where the adverb nearly may refer to the word steady or the word shear. These limitations of the applicability of the shear flow concept will be mentioned again, when all the assumptions are listed together with the system of equations, see Sections II.B.l and II.C.l. Shear Flow Geometries with Open or with Closed Stream Lines

The most important shear flow geometries are shown in Fig. 3; Table V [34-951 lists heat transfer studies on those flow geometries. The flow due

to the relative motion of one of the surfaces is called Couettejlow, while Couette

flow

Poiseuille

Couette flow and

@

0

flow

‘fl

Poiseuille flow superimposed

_.. v g

,

\

0

- positive pressure gradient

____

negative pressure gradient

FIG.3. The main simple shear flow geometries [2]: (a) drag flow in the narrow slit between two parallel plates (plane Couette flow), no pressure gradient; (b) axial drag flow between two coaxial cylinders (annular Couette flow), no pressure gradient; (c) flow through a pipe with constant circular cross section (Poiseuille flow); (d) flow through a narrow slit (Poiseuille flow); (e) axial flow through an annulus (Poiseuille flow); (f) helical flow (flow through an annulus with rotating inner cylinder); (9) axial drag flow in an annulus with nonzero axial pressure gradient; (h) drag flow in the narrow slit between two parallel plates with nonzero pressure gradient; (i) angular drag flow in the annulus between two coaxial cylinders (circular Couette flow), no pressure gradient; (k) flow in a cone-and-plate or in a plate-and-plate viscometer. Geometry a will be referred to as a, if the stream lines are open, or a2 if the stream lines are closed (limiting case K + 1 of geometry i).

TABLE V : HEAT TRANSFER STUDIES IN STEADY SHEARFLOW Temperature field Fully developed

Thermal boundary condition

a1

b

d

C

see a,

34-41

15. 37. 42

mixed

see a2

55E

42

constant wall temperature

60.61

62-64.65E. 66,67E, 68E, 69-71, 72E, 73, 74,75E, 76-79, 133

17,80-82

constant wall temperature

Flow geometries with closed stream lines

Flow geometries with open stream lines e

f

g

k

h

a2

43

31, 38,42, 44-48

38, 42, 45, 49-51E

52, 53E, 54

38, 42. 45, 46, 55E. 56,57E

38,4549, 51E,55E, 58E, 59E

54

i

constant heat flux at wall, adiabatic wall

Developing

constant heat flux at wall, adiabatic

63,67E, 68E, 70, 71, 72E, 76, 78, 79

mixed

90E, 93E. 94E, 95E

77, 83

82, 84, 85

89

83

17,90E,

77,90E,

86E, 87,88

84.85, 91E

89,92

Letters u, b, . . . ,k refer to Fig. 3, where flow geometry a occurs with open and with closed stream lines. The numbers refer to the list of literature. Experimental studies or studies which contain experiments are marked with an E.

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

22 1

the flow due to a pressure gradient is called Poiseuilleflflow.In the literature, most emphasis has been laid on the fully developed temperature field in pipe flow and in Couette flow, and on the developing temperature field in pipes with circular cross section. Experimental studies or studies that contain an experimental part are marked by an E. Poiseuille flow in a pipe with constant but irregular cross section or Poiseuille flow in curved channels with constant cross section induce some small secondary flow in the cross section [96]; this will be mentioned in Section I11 on nonviscometric flow. A detailed description of the historical development of shear flow analysis can be found in the introduction of original papers on the different problems (see for instance [55] for the fully developed temperature profiles and [77, 781 for developing temperature profiles in pipe flow). This study tries to describe the various aspects of shear flow analysis and give credit to the different authors in connection with the arguments in the analysis. Stream lines are lines whose tangents are everywhere parallel to the velocity vectors. In steady flow, the stream lines describe the paths of fluid elements. The heat transfer in the various shear flow geometries depends on whether the stream lines are open (type a,-h in Fig. 3) or closed (type a,, i, k in Fig. 3). In steady flows with open stream lines, the temperature is locally constant with time (dT/dt = 0); for displacements along the flow direction, however, it changes until a fully developed temperature field (DT/Dr = 0 for T , = const) is reached, where conduction and viscous dissipation balance. In processing equipment, the fully developed temperature field is achieved rarely since the flow channels are not long enough and the thermal boundary conditions usually change in the flow direction. Nevertheless, the calculated fully developed temperature field is very useful as a reference state. The degree of development can be estimated from the value of the Graetz number. The unsteady developing temperature ( d T / d r # 0, aT/dz # 0) in flows with open stream lines has been studied very little, possibly because the numerical or experimental techniques are very involved. In shear flows with closed stream lines, the temperature is assumed to be uniform along the stream lines, but locally changing with time (dT/a@= 0; dT/dt # 0) during the starting phase. The stream lines are supposed to be circles, and 0 is the coordinate in the flow direction. Convective heat transfer has no influence on the temperature field. After some developing time, a constant temperature field is reached where viscous dissipation and conduction balance. The degree of development can be estimated from the value of the Fourier number. Drag flow in a narrow slit (plane Couette flow) is introduced twice (al and a, in Table V). One might treat it as an entrance value problem (as in a,) and study the axial development of the temperature field beginning from

222

HOWTH. WINTER

some inlet temperature distribution. On the other hand, one might treat plane Couette flow as a limiting case of circular Couette flow (as in a2); i.e., the stream lines are thought to be closed, and there are no changes in flow direction (d/d@ = 0). The temperature develops with time, beginning with some initial temperature distribution. The fully developed temperature field is the same for both cases. A. THERMAL BOUNDARY CONDITION

The specific heat flux 4 at the boundary is given by the thermal conductivity kRuidof the fluid together with the temperature gradient (aT/dr), in the fluid layer next to the wall 4 = -ktluici(JT/ar)w*

(2.20)

The problems of this section are described in cylindrical coordinates z, r, 0, where I is the coordinate perpendicular to the wall. If the thermal boundaries are not taken to be isothermal ( T , # const), the thermal development in the fluid is connected with the thermal development in the wall. The heat flux at the boundary is determined not only by the conduction to the outside of the channel, but also by the thermal capacity of the wall. Both effects will be analyzed separately in the following. 1. Biot Number and Conduction to Surroundings

If the effect of energy storage in the wall is of no influence, the heat flux at the boundary generally depends on the difference of the temperature level of the experiment to some temperature of the surroundings. In the analysis, the temperature gradient in the fluid layer next to the wall is taken to be proportional to the outer temperature difference (T, - T,); T, is the temperature of the surroundings, and T , is the wall temperature, i.e., the temperature at the boundary between melt and containing wall. The coefficient of proportionality is the Biot number [69,71] of equation (dT/Jr), = Bi(T, - TJh.

(2.21)

Bi is already well known for describing the thermal boundary condition during the heating or cooling of solid bodies (see for instance [97,98]). h is a characteristic length of the flow channel, i.e., the gap width of a slit or the radius of a pipe. Equation (2.21) describes just the radial heat flux in the wall; the axial heat conduction in the wall is neglected in the Biot number. For shear flow applications with closed stream lines, the validity of this assumption has to be verified in each case. However, the assumption seems to be reasonable

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

223

for shear flow applications with open stream lines, where the axial temperature gradient is much smaller than the radial one; during thermal development, the heat flux into the wall changes in the flow direction, but beyond a certain distance from entrance into the channel, these changes are small due to a small axial gradient. The value of Bi for certain applications can be derived from a heat balance for the wall. I n some cases Bi is a function of geometry and of the thermal conductivities only. The heat flux through the wall of a pipe, for instance, can be determined from the inner radius rp of the pipe wall, the wall thickness s, the thermal conductivity kwall,and the inner and outer temperatures T , and T , [98, p. 711. (2.22) On the other hand, the heat flux into the wall is determined by the thermal boundary condition (Eqs. (2.20) and (2.21)) (2.23) Thus, for steady pipe flow with controlled temperature at the outer wall ( T I = T w ;T , = Ts), the Biot number can be calculated by equating Eqs. (2.22) and (2.23): (2.24) Applying this formula to capillary viscometry (rp= 0.5 cm, s = 4.5 cm), one finds values of Bi w 20. Examples for pipe flow with 1 I Bi I 100 are given in Fig. 8 of Section II.B.4. Similarly, the outer Biot number Bi, for annular flow (with ri and r, as the inner and outer radius) would be (2.25) and some examples for pipe extrusion give 1 < Bi, < 10. Figure 4 illustrates the geometrical meaning of Bi. The tangent to the temperature curve T(r/h)at the wall passes through a guide point outside the flow channel; the distance between the guide point and the wall is Bi-', and the ordinate is the surrounding temperature T,. For Bi = 10, for instance, the distance of the guide point from the boundary is 1/10 of the gap width h for annular flow or 1/10 of the radius r, for pipe flow. When the Biot number changes in flow direction, one can visualize this by the appropriate displacement of the guide point.

HORSTH.WINTER

224

FIG.4. Thermal boundary condition for channel flow described by the Biot number Bi and the surrounding temperature T,, i.e., by a guide point outside the channel.

The boundary condition for the temperature field is not known in general. If there are temperature data T,(z) available, they can be used in a numerical program. But often one has to guess these conditions to make an estimate of the temperature profiles possible. Most of the studies shown in Table V prescribe idealized conditions such as: constant wall temperature T,

=

const

Bi, -, - co, Bi, +

or

00;

(guide point at the wall),

constant heat flux at wall (dT/dr),

=

const

or

Bi(T, - T,) = const,

adiabatic wall (aT/dr), = 0

or

Bi = 0; (guide point at infinity).

The use of the Biot number allows one to adopt more realistic thermal boundary conditions, and one goal of further experimental heat transfer studies should be the measurement of Bi in various engineering applications. For the examples shown throughout this analysis, the boundary condition at the wall will be described b y Bi and T, independent of z (for steady flow with open stream lines) or independent oft (for flow with closed stream lines), respectively. If the value of Bi is finite, the wall temperature T,(z) or T,(t) changes according to the development of the temperature field, and it reaches in the fully developed temperature field. a constant value Tw,m

VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS

225

2. Thermal Capacity of the Wall The Biot number is appropriate for describing heat conduction to the surroundings. However, if the wall stores some energy during thermal development, a different boundary condition is needed to describe this effect. The thermal development should be calculated for the fluid and the wall together. This has been done by Powell and Middleman [92] for plane Couette flow with one wall having a finite mass, which absorbs part of the heat generated by viscous dissipation. The thermal development was found to be significantly retarded by the response of the boundary; the parameter characterizing the retardation is the ratio of mass times heat capacity of the solid wall and that of the fluid: (mcp)wa,,/(mcp)R,,d. In this study, however, detailed calculation of the temperature field in the wall will be-avoided by introducing a capacitance parameter C . For flow with closed stream lines, the wall temperature changes with time during the thermal development. The rate of thermal energy stored in the wall is assumed to be proportional to the time change DT,/Dt of the temperature at the boundary. The temperature gradient in the fluid layer near the wall becomes

(2.26) The capacitance parameter C is dimensionless, and the ratio hla,,, is used in Eq. (2.26) because below the whole boundary condition will be made dimensionless. C is determined by the geometry and by the capacitance of both the fluid and the wall. The heat flux to the surroundings is kept proportional to the outer temperature difference T, - T,. Assuming constant thermal properties of the wall material, the rate of energy storage in the wall is proportional to the time change of the average temperature of the whole wall. The time change of the average temperature of the wall might not be proportional to the time change DT,/Dt at the boundary. Thus, the capacitance parameter describes the effect of energy storage in the wall only approximately. In many polymer engineering applications, however, the thermal development in the wall is much faster than the thermal development in the fluid, and the temperature at the boundary is representative for the whole wall. An example where uniform temperature in the wall is assumed will be given in the following. Example for C: The temperature of the inner cylinder of a Couette system (geometry i in Fig. 3, ri = inner radius, r, = outer radius, h = r, - r i ) changes during the thermal development of a shear experiment. The temperature of the inner cylinder is assumed to be uniform and equal to the

HORSTH. WINTER

226

temperature at the boundary; this is justified, if the ratio of the Fourier numbers (see Section II.C.2) is small: (2.27)

Axial heat conduction is neglected and, of course, the Biot number is equal to zero. The heat flux into the inner cylinder is balanced by the temperature raise of the inner cylinder: 2nri(aT/ar)w

kRuid

=

nri2(PC)cylindcr

aTw/at.

(2.28)

By comparing Eq. (2.28) with Eq.(2.26)one finds the capacitance parameter for the Couette system: (2.29) The influence of C on the thermaI development will be shown in Figs. 23 and 24 of Section II.C.3. For most of the steady heat transfer problems with open streamlines, the walls are stationary or just rotating about the z axis. The capacitance of the wall has no influence on the temperature (DT,,,/Dt = 0). An exception would be the inner boundary of axial Couette flow (geometry b or g of Fig. 3) as occurs in the wire coating die. The corresponding thermal boundary condition is vwh aTw Ts - Tw + ci--. = Bi, (2.30) h athid aZ u, is the axial velocity of the wall (inner cylinder). For the example of a wire coating process, a heat balance for the inner cylinder (wire) leads to the same formula for the capacitance parameter as for the Couette system above, Eq. (2.29). The assumptions made were uniform temperature in cross section of wire and no axial conduction in the wire. The Biot number for the wire is zero. The geometrical meaning of the boundary condition with capacity and conduction to the surroundings is shown in Fig. 5. The guide point is not at a constant position (as for flow with DT,/Dt = 0; see Fig. 4),but moves during the thermal development along T, = const. For the fully developed temperature field, the wall temperature does not change any more; the capacitance of the wall is of no influence, and the guide point is, as in Fig. 4, at a distance Bi- from the boundary. An adiabatic or perfectly insulating wall would be a wall without thermal capacitance (or with an appropriate heat source of its own); the corresponding Biot number and the capacitance parameter are both equal to zero.

);r

W

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

227

displacement of guide point during thermal development

FIG.5. Thermal boundary condition for a wall with thermal capacitance.During the thermal development,the guide point of the tangent on the temperature field moves toward the position (Bi- T J .

’,

B. STEADYSHEARFLOWWITH OPENSTREAMLINES Steady shear flow with open stream lines could be analyzed now by going into each of the shear flow geometries al-h in Fig. 3. Instead, it will be demonstrated here that the helical flow geometry is representative since all the other geometries are limiting cases of helical flow. In helical flow, the fluid flows through an annulus between two concentric cylinders (Fig. 6). Axially, the fluid flows due to a pressure gradient and/or due to the axial movement of the inner (or outer) cylinder. In the circumferential direction, the fluid flows due to the rotation of the inner cylinder. Fluid elements move on helical paths; the angle of the helices flow due to pressure gradient /and due to axial movement

path of !bid particle /

/

/

1

//// / / /// / / / /’/ ’,’,.’//,’/,’////

/ / / / ’ i//

FIG.6. Helical flow geometry.

flow due to rotation of inner cylinder,

HORSTH.WINTER

228

depends on the ratio of the axial to the circumferential velocity component, which both depend on the radial position of the fluid element. The annular geometry is characterized by the ratio of the radii IC =

(2.31)

ri/ra,

The limiting cases are the pipe ( I C = 0) and the plane slit ( K tion in the annulus is given by dimensionless coordinates : r - ri ra - ri

radially:

Y

= ___ ,

axially:

2

=

Z ~

1 Gz’

-, 1). The posi-

O I Y I 1 ;

(2.32)

0 I Z I Gz-’.

(2.33)

The value of Z indicates, as will be shown in the following, to what degree the temperature field is developed along the channel. The Graetz number Gz will be defined in Eq. (2.56). 1. Assumptions and System of Equations

The equations of change, Eqs. (1.1)-(1.3), have to be simplified before they can be solved. First the assumptions will be listed, then they will be commented upon: incompressible fluid with constant thermal conductivity and diffusivity ; steady laminar flow (a/& = 0); rotational symmetry (a/a@ = 0 ) ; velocity gradients

no slip at walls; inertia negligible ; kinematically developed velocity at z = 0; gravity negligible; viscosity measured at constant temperatures and constant shear rates gives applicable local values of the viscosity during temperature changes and during small changes in shear rate; rheologically developed stress at z = 0; convective heat transfer much larger than conduction in flow direction; heat transport toward the walls by conduction only. Throughout this section the molten polymer is taken to have constant density ( p / p = 1; E = 0), constant thermal conductivity (k/E = l), and constant thermal diffusivity (a/a = 1). In the general system of equations for helical flow, however, these properties have been kept as variables, and one might

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

229

evaluate them in the numerical program using p ( p , T ) , Up, T ) ,and u(p, T ) data from measurements on the fluid at rest. The density is assumed to be constant since in actual experiments (with T # const, p # const) density changes are delayed, i.e., the changes are overestimated if one applies p data of equilibrium thermodynamics. For this reason, the effect of expansion cooling is not considered in all but one example. For the one exception, the supposed expansion cooling term (containing E T )of Eq. (1.8) is kept in brackets in the energy equation; the effect of expansion cooling is estimated in an example of pipe flow; see Fig. 13 with E # 0. During the axial development of the temperature field, the temperaturedependent viscosity is changing and causes the shape of the velocity profiles to change accordingly. For continuity reasons, the changes of the axial velocity require some radial flow. Using the equation of continuity, Eq. (l.l), the radial velocity components have been estimated from the change of the lo-’ ij). axial velocity component and have been found to be small (lo,( Throughout Section ILB, the influence of the radial velocity components on the radial heat transfer, on dissipation, and on the viscosity will therefore be neglected. Due to rotational symmetry and due to du,/dr and r a(u,/r)/dr being the largest gradients, the isotropic pressure is taken to be a function of z only:

-=

P

=

P(4.

The shear rate becomes

(2.34) which is the root of the second invariant of the rate of strain tensor. In most applications, molten polymers do not slip at the wall, and in all the published heat transfer studies this assumption has been made. Polymeric materials such as high density polyethylene, polyvinylchloride, or polybutadiene, however, seem to slip in certain ranges of the normal stress and shear stress at the wall [99,100]; the velocity field is then drastically changed and additional frictional heating occurs on the sliding surfaces. The velocity at the entrance ( z = 0) is assumed to be fully developed; i.e., inertial effects are neglected, and the stress is assumed to be governed by the three viscometric functions (of steady unidirectional shear flow) at the local shear rate and the local temperature. For low Reynolds number pipe flow of inelastic liquids, the kinematic development is practically completed after a length of 1 = O.lra Re [loll. Neglecting inertia might therefore be justified for entrance flow of molten polymers, which is low Reynolds number flow.

230

HORSTH. WINTER

The rheological properties of the polymer entering the annulus are determined by a flow and temperature history, which obviously is different from that for steady shear flow. Judging from the measured pressure profiles along a slit die, steady shear flow might be reached practically at l/h = 20-30 (depending on geometry, flow rate, and polymer melt). Therefore, the heat transfer study of this section may give unrealistic results for flow in short annuli and short circular holes, which rheologically should be treated as an entrance flow problem. The equations for conservation of mass, momentum, and energy in helical flow are

(2.35)

The average axial velocity is u, =

2 ra2 - riz

u,r dr.

(2.36)

The initial and the boundary conditions are T(r,0) = T,(r)

(2.37)

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

23 1

The meanings of Bi and T , have been described already in Section 1I.A on the thermal boundary condition (Bi, 0). The capacitance parameter C of Eq. (2.30)has been omitted here since the temperatures are assumed to be steady and since the walls are stationary for most shear flow applications with open stream lines. For the dimensionless presentation of the equations, a reference velocity B can be defined by vector addition of the mean axial velocity b, and the mean circumferential velocity ~ , , ~ / 2 : -

v =

[vz2 + (VeJ2) 2 ] 112 .

(2.38)

The reference length is taken to be the gap width

h = r, - r i . (2.39) For pipe flow the reference length becomes equal to the pipe radius (h = ra). Using the reference velocity 3 and the reference length h, one can define a reference shear rate p = b/h (2.40) and a reference viscosity (2.41) rl = v(74 To). T o is a characteristic temperature level of the experiment, for instance the average melt temperature at the inlet ( T o= Te).Flow problems with viscous dissipation do not have a characteristic temperature difference to which temperature changes can be related. Some authors relate the temperature to the temperature level T o ;this, however, seems to be rather arbitrary since T / T o might assume different values in otherwise similar processes (e.g., at different temperature levels T oand To').The value of T / T oadditionally depends on the choice of temperature scale. Therefore, the temperature coefficient of the most temperature-sensitive property, the viscosity, has been used to define the dimensionless temperature: (AT)ref= 0-l. The dimensionless variables are:

velocity

(vR>

v@,

=

( v ~ / u@/B, ~ , uz/D)?

(2.42)

pressure gradient

(2.43)

shear stress

(2.44) (2.45)

radial position (2.46)

232

HORSTH . WINTER

viscosity

temperature The dimensionless form of the equations becomes

(2.49) (2.50) (2.51 )

The dimensionless average axial velocity is

VZ =

2

1

VzR dR,

(2.53)

(2.54)

VISCOUS

DISSIPATION IN FLOWING MOLTENPOLYMERS

233

In the expansion cooling term of the energy equation, the absolute temperature T is not replaced by the dimensionless temperature 9 since the dimensionless product ET can be considered as constant within the accuracy of the calculations. The system of equations is formulated in cylindrical coordinates, and R = r/ra is kept as a dimensionless coordinate, even if h = r, - ri is the reference length and not rs. The results are presented using the dimensionless coordinate Y . If one wants to avoid this inconsistency, the substitution

R = Y(l -

K)

+ K,

dR = dY(1 -

K)

eliminates R from the equations; with this substitution the equations look unnecessarily complicated and therefore both coordinates are kept : R in the equations and Y in the graphical presentation of the results. For pipe flow, R and Y are identical.

2. Dimensionless Parameters The problem as stated in Eqs. (2.49)-(2.54) is completely determined by six dimensionlessparameters (Na, Gz, K , m, Vz,L ) together with the boundary conditions. A general description of the dimensionless parameters has been given by Pearson [3]. If the pressure dependence of the viscosity or nonconstant thermal properties would be included, the number of parameters would increase accordingly. The equation of motion and the equation of energy, Eqs. (2.35),are coupled by the temperature-dependent viscosity. The extent of the coupling increases with the value of the Nahme number [44] : Na

=

j02q/X

(2.55)

which compares the dissipation term with the conduction term in the equation of energy. For values of Na greater than 0.1-0.5 (depending on geometry and thermal boundary conditions), the viscous dissipation leads to significant viscosity changes, i.e., changes reflected in the T and u fields. For smaller values of Na, isothermal conditions can be achieved practically; in this case, the equation of motion can be integrated independently of the energy equation. In some studies the Brinkman number [62] Br = B2ij/kTo has been used instead of the Nahme number. However, Br contains the arbitrary temperature level To(since no characteristic temperature difference is available) and may, therefore, have very different values for similar processes. The value of Br does not give any information on the extent of the coupling between the equation of motion and the energy equation. (Note that the Nahme number sometimes is called the GrBith number after Griffith [102], who used the same dimensionless group in one of the later applications.)

HORSTH. WINTER

234

The energy equation contains a convection, a conduction, and a dissipation term. By comparing the convection and the conduction terms one arrives at the Graetz number [1031

(2.56)

GZ = Vzh2Jzil

which has been included in the dimensionless form of the z coordinate. The Graetz number can be understood to be the ratio of the time required for heat conduction from the center of the channel to the wall and the average residence time in the channel [75]. A large value of Gz means that heat convection in flow direction is more important than conduction toward the walls, Gz = 100, for instance, is a common value for extruder dies. (Note that some authors define the Graetz number Gz n.) The Gz number has been defined with the average axial velocity and the length of the annulus, instead of the reference velocity ij and some mean path length Tfor the fluid elements in the annular section. One might, however, define Tto be T = 1ijpzwhich would result in a Graetz number Gz = ijh2/uTequal to the one defined in Eq. (2.56). The value of the dimensionless average axial velocity

Y.=B=[I+(gJ]

2

-1/2

V

,

O
(2.57)

describes whether the flow tends to be closer to axial flow in an annulus

(Vz= 1) or closer to circular Couette flow (Vz= 0). The dimensionless length of the annulus is

L

=

l/h

=

l/(ra -

Ti).

(2.58)

Another dimensionless parameter originates from the shear dependence of the viscosity: the power law exponent m in Eq. (2.47).

3. Universal Numerical Shear Flow Program The system of equations is solved by an iterative implicit method (aJaZ described by a backward difference; alaR and a2faR2 described by center differences;gradients at a boundary are calculated from a parabola through three points), similar to the one used in an earlier study on helical flow [85]. A network is superimposed on the annulus. Difference equations are then derived for each node point, which fulfill the condition of conservations of mass, momentum, and energy. The method described in the following was found to converge rapidly; for example, a run with 60 radial and 250 axial steps requires a computation time of about 30 s. The solution procedure is an iterative one, in which the coupled equations are linearized and solved separately. The nonlinear terms and the coupling

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

235

conditions have to be satisfied by alternating improvements on the velocity and on the temperature field. The iteration is terminated when the relative change in successive steps becomes smaller than one thousandth ( A ~< ~ ,1 0 - 3 ) . The flow chart in Fig. 7 describes the structure of the program. The velocity field at the enrrunce, which is assumed to be fully developed kinematically, is calculated by taking, as a first guess, a Newtonian viscosity according to the entrance temperature field. The shear dependence of viscosity is included then by iteration, using the improved values of the velocity field. After about 6-20 iterations, the velocities reach values that are practically constant. geometry. material and process data, boundary conditions Bi, , Bi., G.,, c,o ve,, , v,,, ; entrance conditions T (r); z=o; t,=l, v e z v,= 0 .

.

dimensionless parameters No , Gz,Vz ,x , dR = const. logarithmic steps sizes AZ, , number of steps no , n -- 0 I

conditions

;

4,. 0

end

FIG.7. Flow chart of universal shear flow program.

236

HORSTH. WINTER

The axial velocity V, and the pressure gradient P' are calculated from the Z component of the equation of motion together with the integral of the flow rate (Eqs. (2.51) and (2.53)). The circular flow V, is evaluated from the 0 component of the equation of motion (Eq. (2.50)).The R component V, of the velocity supposedly does not influence the viscosity and the convection; thus it is calculated separately at the end of each step using the equation of continuity (a numerical method that allows for positive or negative radial flow contributions has been suggested by Gosman et al. [104]). The entrance conditions are then stored, and the fully developed temperature jield is calculated so as to be available as a reference state for the developing temperature field. In the fully developed temperature field, the convective term of the energy equation is zero. The iteration starts out with the viscosities and velocities at the entrance. They give a first approximation of the dissipation term and of the fully developed temperature field. Using this solution, one gets improved values of the viscosities and the velocities by iteration. These values of the viscosities and velocities lead to the second approximation of the fully developed temperature field, and so on. After the values of (9, - ~ ( I c00)) , and satisfying the condition of (A,,J < of (9, - 9(1, 00)) are stored as reference values for the developing temperature field. Then the program goes back to the entrance temperature field and starts calculating the developing temperature, velocity, shear stress, and pressure. If both walls are adiabatic (Bi, = Bi, = 0), there does not exist a fully developed temperature field; the program then starts calculating the developing temperatures immediately. For flow in a capillary (IC= 0), the velocity and the temperature have a zero gradient at R = 0. The power law model fails in describing the viscosity at low shear rates, and for computational purposes at least one has to set an upper limiting value of the viscosity (this has been done in the numerical program of this study). For more accurate calculations, one has to approximate ranges of the viscosity curve by several power law and temperature coefficients. Also a viscosity table could be used instead. The numerical program has been checked with analytical solutions of the fully developed temperature and velocity field in plane Couette flow and with isothermal flow in a pipe and in an annulus [lOS]. Besides helical flow with its steady, but developing temperature field, the system of equations, Eqs. (2.35)-(2.37), describes the flow geometries of all other steady shear flows with open stream lines (type a,-h in Table V). Therefore, it actually is possible to use one numerical program for all these flow cases. The appropriate values of the ratio of the radii IC,the axial velocity of the inner cylinder V,(IC,Z ) , and the average axial velocity Y' are listed in Table VI.

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

237

TABLE VI SHEAR

FLOW GEOMETRIES AS LIMITING CASES OF HELICAL FLOW"

Flow geometry as described in Fig. 3

u

V.(JG z)

VZ

a1 b

0.999 o
2 determined by iteration for P = 0

1

0 0.999 O < X < l O
-

1

0 0 0

1 1

finite finite

0.999 O
0

a2 1

0

1

O
0

Listing of the corresponding geometry (described with K ) and kinematics (described with the velocity of the inner cylinder I/Z(K.Z)and the average axial velocity Vz).Geometries a,-h are with open, and geometries a2 and i with closed stream lines.

Due to the parabolical character of the solution procedure for the equation of energy, the helical flow program can be applied only to flows with nonnegative velocity components. Thus, axial drag flow in an annulus with nonzero axial pressure gradient (type g) and drag flow in a narrow slit between two parallel plates with nonzero pressure gradient (type h) can be analyzed only up to moderate positive pressure gradients. A solution procedure that allows for back flow is described in the literature [104], but it does not seem to have been applied to these types of flow. 4. Calculated Results There is a large variety ofheat transfer problems solvable with the universal shear flow program. Some examples follow, mainly concerning the thermal boundary conditions (Biot number) and the kinematics for various shear flow geometries. Similar examples of helical flow or annular flow calculations have already been published, however, with idealized thermal boundary conditions [85]. In all the examples of this section, the entrance temperature (at 2 = 0) is taken to be 9,(R) = 0. The thermal boundary conditions influence the developing temperatures and velocities to a large extent. In analytical studies generally, idealized conditions are assumed, i.e., isothermal or adiabatical wall; in real flow

HORSTH. WINTER

238 I

=m

2

No.5 Bi =lo0

rn = 2 . 5

rn =2.5

31R.Z

1

0.5

A

.?=lo-’

z 40”

C

R

0,5-

R

a5

R

1

1

Ro.8. Influence of the thermal boundary condition, described by a guide point outside the channel given by Bi and 9, = 0, on the developing temperature field in pipe flow. Bi = 100 is close to the isothermal wall condition, while Bi = 1causeslarge changes of the wall temperature.

VISCOUS bISSIPATION IN

FLOWING MOLTENPOLYMERS

239

situations the thermal boundary condition is somewhere between the two. The strength of the Riot number in describing a more realistic kind of boundary condition will be demonstrated on pipe flow: the fluid supposedly enters the pipe at a constant temperature equal to the temperature of the surroundings (9, = 9, = 0). Due to viscous Pissipation, the temperatures increase in flow direction (Fig. 8). For Bi = 100, the wall temperature stays nearly constant, while for Bi = 10 and Bi = 1, the wall temperatures already increase in early development. The fully developed temperature at Bi = 10 has a value between that at Bi = 100 and Bi = 1. At large Bi the temperature in the layer near the wall is kept low; the viscosity and hence the viscous dissipation is large, and a large temperature gradient is needed to conduct away all newly dissipated energy. At small Bi the wall temperatures have to rise signifieantly before the heat flux at the wall can balance dissipation: the temperature gradient can still be relatively small since the viscosity and hence the viscous dissipation become small at high temperatures. At large Bi the temperathreh are high because viscous dissipation is most pronounced. At small Bi the temperatures are high because the conduction toward the surroundings requires large wall temperatures. For intermediate Bi, the fully developed temperature has a minimum. The corresponding pressure gradient P ( Z ) decreases due to the decrease of the temperature dependent viscosity (Fig. 9). The decrease of P'(Z1 below its value at the entrance P(0) is most pronounced at small values of the Biot number, where the wall temperatures increase the most. The temperature gradient at the wall is defined with the Biot number and an outer temperature difference 9, - 9, (see Eq. (2.54)),where $,(Z) itself depends, besides the other parameters, on Bi. The dimensionlesstemperature

FIG.9. Pressure gradient in pipe flow at different thermal boundary conditions; decrease due to the thermal development described in the previous figure. The pressure gradient for isothermal pipe flow can be calculated analytically: P(0) = - 2 ( m + 3)"".

240

HORSTH. WINTER

gradient [as(&2)/dRlw increases with Z till it reaches its fully developed value at about Z = 1 (see Fig. 8).

a. Nusselt Number. For engineering calculations, the specific heat flux q often is described by means of the Nusselt number [lo61 (2.59) and a characteristic temperature difference (AT)ref.The product Nu k/h sometimes is called the heat transfer coefficient. The value of the Nusselt number depends much on the choice of the temperature difference ( A T ) r e f . For flow without signijcant viscous dissipation, one takes it to be the local average temperature difference ( T , - T ( Z ) ) or the average temperature change ( T ( Z ) - T(0))in the flow direction. The average temperature of the melt is chosen to be the “cup mixing temperature”

T ( Z ) = ___ 1 - uz

T ( R , Z)Vz(R, Z ) R dR

(2.60)

which would be the temperature of the homogeneous fluid after mixing (constant specific heat per volume assumed). The Nusselt number in its usual definition is not adequute for describing the wall heat flux in flows with signijicant viscous dissipation. One disadvantage of the use of Nu is the fact that both Nu@) and T ( Z )have to be known to calculate the wall heat flux q ( Z ) ;the main disadvantage, however, is that in Nu an attempt is made to describe two fairly unrelated quantities as a function of each other, the temperature gradient at the wall and the average temperature difference (Tw(Z)- T ( Z ) ) .This will be explained in the following, using pipe flow as an example. Figure 10 shows developing temperature profiles in pipe flow, on the left-hand side with negligible viscous dissipation (Na = 0.001) and on the right-hand side with significant viscous dissipation (Na = 1). The wall temperature is above the entrance temperature (9, = 0; 9, = 0.1); developing temperatures at constant wall temperature (Bi = 00) are drawn as solid lines, while for a thermal boundary condition with Bi = 10 dashed lines are.used. For flow without signijcant dissipation (Na = 0.001), the temperature of the fluid gradually approaches the wall temperature; the temperature gradient at the wall (shown in Fig. 11) decreases monotonically till it becomes zero. However, if there is significant viscous dissipation (Na = l),the temperature in the layer next to the wall increases drastically even before the average temperature is changed much. The temperature gradient changes its sign; the fluid heats the wall, even if the average fluid temperature is below the

VISCOUS

DISSIPATION IN FLOWING MOLTENPOLYMERS

24 1

9IR.ZI

0.:

~

0.2

----- Bi =10 -Bi

=w

S(R, 3, = 0.1

01

O!

z=10-’

0.5

R

1

C

FIG.10. Calculated temperatures in pipe flow with wall temperatures above the entrance temperature. Comparison of the thermal development without (Na = 0.001) and with (Na = 1) significant viscous dissipation. The wall temperature is taken to be constant (Bi = a)or described by a guide point with Bi = 10.

wall temperature. This corresponds to a negative Nusselt number or a negative heat transfer coefficient, which is an unrealistic result. The dimensionless average temperature difference (Fig. 12)

S ( Z ) - 3,(2)= p ( T ( Z ) - T , ( Z ) ) = B A T

(2.61)

is negative at the entrance of the pipe (prescribed initial condition) and at least for Bi = co, it increases monotonically with Z . For Na = 0.001, the fluid approaches the wall temperature (9, = LJ,+,); for Na = 1, the average temperature difference goes through zero and approaches a constant value greater than zero. The Nusselt number Nu(Z), which conventionally is defined with this temperature difference, has a singularity when the average

242

HORSTH.WINTER

FIG.11. Wall temperature gradients for pipe flow with viscous dissipation (Na = 1) and without (Na = 0.001).The wall temperature is above the entrance temperature.

FIG.12. Development of the difference between the average temperature and the wall temperature in pipe flow with viscous dissipation (Na = 1) and without (Na = 0.001); the wall temperature is above the entrance temperature.

VISCOUSDISSIPATION IN FLOWINGMOLTEN POLYMERS

243

temperature difference AT is equal to zero [69,76,82]. The specific heat flux at the wall obviously is finite (see the wall temperature gradient in Fig. 11). This seeming singularity (at Z of AT = 0) suggests that the usual choice of the reference temperature difference A T cannot be applied meaningfully to flow problems with viscous dissipation [69]. The same argument is valid if, instead of a constant wall temperature, a guide point is chosen outside the channel, see Figs. 10-12 with Bi = 10. In the definition of the Nahme number (Eq. (2.55)), p-’ is taken to be a characteristic temperature difference, and for defining a Nusselt number for flow with viscous dissipation one similarly may take (ATLeF = B-’.

(2.62)

Introducing this reference temperature difference into Eq. (2.59),the Nusselt number becomes Nu = hg(dT/dr), = (ds/aY),, (2.63) and it then is identical with the temperature gradient at the wall. Equation (2.63) seems to be an adequate definition of Nu for flow with viscous dissipation. The relation between Nu and Bi is given together with Eq. (2.54).For fluids with practically temperature-independent properties, an adequate definition of the Nusselt number for dissipative flow can be made by means of the “recovery temperature” [98, p. 4171. b. Expansion Cooling. In steady flow of compressible fluids with nonzero pressure gradient, the density will change in the flow direction; the equation of energy, Eq. (1.7),contains a term that describes the cooling or heating due to those density changes cT DplDt, where r T can be determined from Eq. (1.10). In the example of Fig. 13 the influence of expansion cooling is shown on developing temperature profiles in pipe flow (with 9, = 9, = 0). Values of ET = 0, 0.1, 0.2, 0.3 have been used since values of this magnitude can be evaluated from equilibrium thermodynamic data of molten polymers at rest. The applicability of equilibrium data to regions of rapid pressure changes is still an open question. c. Thermal Development. For constant inlet temperature equal to the temperature of the surroundings, the average temperature increases during the development of the temperature field with increasing Z. The temperature is fully developed at 2 = 0.5-2. Although the absolute value of the average temperature depends on the power law exponent m, or Na, and on Bi, the relative development is nearly the same for the different examples of pipe

244

HORSTH. WINTER

i

0

0.5

R

FIG.13. Calculated temperature profiles in a pipe with constant wall temperature equal to the entrance temperature. The magnitude of ET determines the amount of cooling due to expansion with decreasing pressure p; T is the absolute temperature.

flow (Fig. 14).In annular flow, however, the relative thermal development depends on the thermal boundary conditions (Fig. 15); if the dissipated heat can be conducted to both walls (Bi, = -10'; Bi, = lo5), the developing length is much smaller than if one wall is nearly adiabatical (Bi, = -1; Bi, = lo5).For Newtonian fluids (m = l), the channel length required for thermal development is shorter than for fluids with shear dependent viscosity (m = 3; rn = 5, for instance).

d . Zero Pressure Gradient or Zero Wall Shear Stress. The versatility of the program will be demonstrated on some velocity profiles of isothermal flow in an annulus, including the limiting cases of a plane slit (K:+ 1) and pipe (K = 0). Annular flow with zero velocity gradient at the inner wall (which also means zero shear stress at the inner wall) can be achieved with the appropriate pressure gradient and the appropriate velocity V'(K) at the inner wall (Fig. 16). An application of this type of velocity field could be die flow in the wire coating process: operating at low shear stress at the surface of the wire prevents ruptures of the wire. If the pressure gradient in the annulus

VISCOUS DISSIPATION IN FLOWING MOLTEN POLYMERS

245

1 I

10-2

1o-~

1o-z

axial

10-1 position

axial

7

10” position

7

1

10

1

10

FIG.14. Development of the average temperature in pipe flow at different Bi and m ;(a) at Na = 1 and (b) at Na = 5.

axial position Z

FIG.15. Development of the average temperature in annular Poiseuille flow at different Bi,, Na, and m. K = 0.5; Q = 0.

HOWTH. WINTER

246

9 I1

x

1.0

-

z

->aB -._ 0

0

2 0.5 -

0

0

as

1

radial position Y

FIG.16. Calculated velocities and pressure gradients for isothermal axial flow in an annulus with zero shear stress at inner wall. Parameter is the ratio of radii K.

is prescribed to be zero (P' = 0), the axial velocity of the inner cylinder V'(K) has to be the larger, the smaller IC is (Fig. 17); for a plane slit (K + l), the velocity gradient is constant and the velocity of the moving wall is twice the average velocity, obviously. Taking different values V,(K) in an annulus of K = 0.4 (Fig. 18), the pressure gradient P' adopts positive or negative values. A zero shear stress at the inner wall or a zero pressure gradient at isothermal flow does not mean that this condition applies to the whole flow channel: due to the thermal development the velocity changes, and accordingly a nonzero shear stress at the inner wall or a nonzero pressure gradient arises. 5 . Experimental Studies

The main motivations for undertaking experimental studies on heat transfer in steady shear flow with open stream lines seem to be: investigatingthe validity of the assumptions made in the analytical studies; information on the thermal boundary conditions, i.e., values of Bi in different applications. The flow geometries chosen for experiments were pipe flow and helical flow (see Table V). The measurable quantities were the flow rate, the pressure

VISCOUSDISSIPATION IN FLOWING MOLTEN POLYMERS

radial position

247

'k

FIG.17. Calculated isothermal velocity profiles in an annulus at zero pressure gradient. Parameter is the ratio of radii K.

"0

0.5

radial position Y

1

FIG.18. Calculated isothermal velocity profiles in an annulus of K = 0.4. Depending on the values Vz,i,the pressure gradient P' adopts positive or negative values.

248

HORSTH. WINTER

profile, the radial temperature distribution at the inlet and at the exit, the thermal boundary conditions ; additionally, for helical flow one could measure the torque and the angular velocity of the cylinders. As input data for the numerical program one needs the flow rate (or a pressure gradient), the properties of the polymer (viscosity q(P, T ) ,thermal diffusivity a(T),thermal conductivity k(T),density p(p, T ) ) ,the melt temperature at the inlet Te(r), and two boundary conditions each for the temperature and the velocity fields. The other data can be used for a check on the validity of the numerical solution. Several of the published experimental studies do not specify the data needed for comparing with analytical solutions. While wall temperatures can be measured quite accurately, the temperature measurements in the flowing molten polymer always contain some systematic errors. A thermocouple mounted on the tip of a probe is placed into the melt stream. The probe is supposed to adopt the melt temperature as closely as possible (zero temperature gradient in the polymer layer next to the probe). Apart from distorting the velocity profile by introducing the probe into the flow, two effects are influencing the temperature measurement: heat conduction along the probe, which requires a heat flux and a temperature gradient in the polymer layer next to the wall of the probe, and viscous dissipation in the polymer around the probe. The error due to conduction along the probe can be excluded by setting the base temperature, where the probe is mounted to the wall of the channel, equal to the temperature at the tip of the probe [107,108]. The error due to dissipation cannot be avoided, but it can be kept small by measuring the melt temperature at positions of very low velocities, i.e., after slowing down the flow in a wide channel and then calculating back to the corresponding temperatures at the exit of the narrow channel by means of the stream function [91,109]. Van Leeuwen [110] studied the applicability ofdifferent probe geometries and found that a probe that is directed upstream parallel to the streamlines of the undisturbed flow gives the most accurate temperature data of the melt. Gerrard et al. [67] pumped a Newtonian fluid (oil) through a narrow capillary ( I , = 0.425 mm and 0.208 mm, 33 I l/r, I 459). They measured the pressure drop, the flow rate, the inlet temperature, the wall temperatures, and the radial temperature distribution at the exit. The calculated values of the pressure drop and the temperature at the exit reportedly agree with the measured values within 5%. The viscosity was taken to be a function of temperature; expansion cooling was neglected in the analysis. Mennig [72] extruded polymer melt (low density polyethylene) through a capillary ( I , = 3.5 mm, l/r, = 225.7) at adiabatic wall conditions. Measured quantities were the temperature in the center of the entering polymer stream, the wall temperature distribution, the radial temperature distribution at the

VISCOUS

DISSIPATION IN FLOWING MOLTENPOLYMERS

249

exit, and the total pressure drop. The calculated values of the wall temperatures and of the radial temperature distribution exceeded the measured values by about 5%. The viscosity has been taken to be a function of shear rate and of temperature; expansion cooling has been included in the analysis. For capillary flow, Daryanani et al. [75] measured the average heat flux through the wall using an electrical compensation method. From the total pressure drop and the heat flux through the wall, they calculated the average temperature increase between entrance and exit of the capillary. Winter [91] extruded a polymer melt (low density polyethylene) through an annulus ( K = 0.955 and 0.972) with rotating inner cylinder. The measured quantities were the mass flow rate, the pressure distribution, the rotational speed of the inner cylinder, the radial temperature distribution at the entrance and at the exit, four temperatures each at the inner and at the outer wall. As shown in Fig. 19 the developing temperatures have been calculated beginning with the measured temperature distribution at the inlet. For the exit temperature distribution, measured and calculated values agreed up to Y zz 0.75 within 5% of the temperature increase (at the outer wall, 0.75 IY 5 1 the temperature distribution has not been desribed sufficiently with only four temperature readings). The measured and calculated pressure gradients agree within 8%. Expansion cooling has been neglected in the analysis.

FIG.19. Comparison of measured and calculated temperature profiles in helical flow [91].

250

HORSTH. WINTER

C. SHEARFLOWWITH CLOSEDSTREAM LINES The shear flow geometries with closed stream lines studied most widely are circular Couette flow and its limiting case, i.e., plane Couette flow (IC+ 1). The fluid is sheared in the annular gap between two concentric cylinders in relative rotation to each other (Fig. 20). The axial velocity component u, is zero. At time t = 0, the Couette system is started from rest at isothermal conditions with a step in shear rate (Q(t I 0) = 0 and Q(0< t ) = fo = const); alternatively the system might be started with a step in shear stress. Three types of development are superimposed on each other, each of them on a different time scale: Kinematic development: The fluid has to be accelerated until it reaches a velocity and a shear rate independent of time. The kinetic development can be calculated for a Newtonian fluid; a practically constant velocity field is reached after [1111 t

=

ph2/16q

(2.64)

(h is the gap width, q the constant Newtonian viscosity, and p the density).

For viscoelastic liquids an estimate on the duration of kinematic develop-

FIG.20. Flow geometry of circular Couette flow.

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

25 1

ment can be made from the loss and the storage modu1esG”andG’measured in periodic shear experiments at frequency w = l / t [1121 t

>> ~ ( P / G ” ) ” ~ or

t >> h(~/G’)l’~

(2.65)

For startup experiments on polymer melts, the kinematic development generally is assumed to be completed before the rheological and the thermal development has actually started. Rheological development: The viscosity ~ ( 9T, , t) needs some time of deformation at constant shear rate, until it adopts a constant value.

Thermal development: Due to viscous dissipation beginning at time t = 0, the temperatures in the gap rise until the temperature gradients toward the walls are large enough to conduct away all the newly dissipated energy. Convection does not influencethe temperature field because the temperatures along stream lines are constant. 1. Assumptions and System of Equations

The assumptions corresponding tothe ones listed in Section II.B.l are: incompressiblefluid with constant thermal conductivity and diffusivity ; no change in z direction; rotational symmetry (a/a@ = 0); velocity # 0;U, = v, = 0; no slip at walls; inertia negligible; kinematically developed velocity at t = 0; gravity negligible; viscosity measured at constant temperatures and constant shear rate gives applicable instantaneous values of the viscosity during temperature changes and during small changes in shear rate; rheologically developed stress at t = 0.

The stress‘equation of motion and the energy equation become (2.66) (2.67)

The reference values are chosen to be the same as in the helical flow analysis: v = veVi/2; h = ra - ri; 9 = ij/h; i j = ~ 6T o,) .

252

HORSTH . WINTER

The dimensionless variables are velocity

v,

radial position

R = r/ra = (1 - K ) r / h ,

=

y=shear stress

PR8

=

v,P, K

IR I 1,

r - ri , 05Y11, r, - ri

h Gc3==3

uvl

9 = P(T - TO).

temperature

The dimensionless form of the system of equation reads (2.68)

as

-pc

pE aFo

= (1

-

K ) ~ [ & (asR +~ ~ ) k

Na;(Rsxr]. a

v,

(2.69)

The initial conditions are

w,0) = 0,

V,(K 0) = [email protected](R) (2.70) where V',JR) is the kinematically developed velocity at the initial temperature. The boundary conditions are d 9 ( ~ Fo) , ss,i- ~ ( I cFo) , = Bii dR 1--K

1

( ~Fo) , +-1 Ci- K d 9dFo

a q i , FO) 9,, - 9(1, Fo) -- C, d9(1, Fo) = Bi, aR 1-u 1 - K dFo

0 = Fo. (2.71)

V@(K, Fo) = 2 Ve(1, Fo) = 0

The thermal boundary condition is an energy balance of the inner and of the outer wall. The heat flux into the wall is equal to the heat flux out of the wall minus the change of energy stored in the wall. The boundary condition has already been described in Section 1I.A. It is repeated here to show the complete mathematical problem at once (Bi, < 0; Bi,, Ci, C, > 0). 2. Dimensionless Parameters For a description of most of the dimensionless parameters, the reader is referred to Section II.B.2. The Nahme number, Eq. (2.55), compares the dissipation term and the conduction term of the equation of energy. The

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

253

ratio of radii K shows the influence of curvature, and m describes the shear thinning effect of the viscosity. Instead of the Graetz number one introduces as a dimensionless variable the Fourier number

FO = ta/h2

(2.72)

which can be understood as the ratio of the current time of the experiment and the time needed for heat conduction from the center of the channel to the wall. At Fo = 1-4, depending on the thermal boundary conditions, the thermal development is completed. The Fourier number corresponds to Z in the heat transfer problem with open stream lines, where one might define an average residence time f = z/O,:

Z

=

Z

1

za ai = -= h2 = Fo.

h2a,

(2.73)

3. Solution Procedure and Calculated Results The 0 componentLof the equation of motion is the same for annular shear flow with open and with closed stream lines. The kinematically developed velocity V,( Y, 0) at isothermal conditions can be calculated with the existing numerical program of Section 1I.B without any changes. The same is true for the thermally developed case at large times at constant thermal boundary conditions since the conduction and the convection terms are identical in both types of flow. If one replaces Z by Fo and sets V,(Y) = pz = (which is an arbitrary small value to avoid singularities in the program), even the developing velocity V@(Y,Fo), temperature S(Y, Fo), and shear stress PRe( Y , Fo) can formally be taken from the existing program without further considerations; see Table VI. The capacitance parameter, however, has to be included in the thermal boundary condition. The solution procedure is basically the same for steady shear flow with open stream lines and for unsteady shear flow with closed stream lines (Couette system), and it would have been possible to treat it in one special section in the beginning. For two reasons, however, this has not been done in this study: (1) shear flow with open stream lines is much more important for polymer processing; (2) the frequent change from Z to Fo would make the explanations difficult to comprehend. The solution procedure in Section 1I.B is meant to be an example, and it will not be described repeatedly for the corresponding problem in this section. The geometry of a cone-and-plate or a plate-and-plate viscometer cannot be described by the existing shear flow program. Turian and Bird [52-541 estimated the temperature effects in cone-and-plate systems by applying the maximal gap width (at the outer radius) to a plane Couette system with

HORSTH. WINTER

254

isothermal walls. The radial heat conduction, which might diminish the effect of dissipation, is neglected. The development of the temperatures in circular Couette flow is a function of the dimensionless parameters Na,Fo, IC,m,and of the thermal boundary conditions. In Figs. 21 and 22, the influence of the geometry on the developplane slit

- 0.8 -

I Y = 0.999)

9

>. -

e

>.

.

annulus with x = 0.5

\ \< \

-

* .

6

\

\

0 . A\\

\ \\ \

\

\

FIG.21. Comparison of developing temperatures for plane and for circular Couette flow. The outer wall is taken to be isothermal; the inner wall is close to isothermal (Bii = - 100) and close to adiabatical (Bi, = - 1). Na = 1; rn = 2; C, = 0.

I4 > 9

l%-1

2

e

c

E

2

(u

ff

0.5

6 W z

c

0 -

P O

10-~

lo-*

lo-'

1

dimensionless time Fo

FIG.22. Development of the average temperature in plane and in circular Couette flow. The solid lines correspond to the development with both walls close to isothermal (Bi, = - 1001, and for the dashed lines the inner wall has been taken to be close to adiabatical (Bii = - 1). Na = 1 ; m = 2;C, = 0.

VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS 255 ing temperature 9(Y, Fo) will be demonstrated for plane Couette flow 1) and for circular Couette flow with K = 0.5, both with constant temperature of the surroundings equal to the initial temperature (9(Y, 0) = gS,,= 9,,, = 0); the outer wall is taken to be isothermal (Bi, = co) and the inner wall is taken to be close to isothermal (Bi, = -100) and close to adiabatic (Bi, = -l), respectively. The thermal capacitance of the wall is neglected (Ci = 0). For the plane slit the shear rate and hence the viscous dissipation are nearly uniform. The temperatures rise uniformly until the conduction toward the walls takes more and more heat out of the channel. When the temperature gradients are large enough to conduct away all the newly dissipated energy, the fully developed temperature field is reach. If the inner wall is nearly adiabatical (Bi, = - l), the temperature gradient has to adopt larger values since nearly all the dissipated energy has to be conducted to the other wall on the outside. The corresponding temperatures for circular Couettepow ( K = 0.5) are asymmetrical through the geometry of the system, additionally to the asymmetry of the thermal boundary condition. The shear rate and the viscous dissipation is much larger at the inner wall than at the outer one. The comparison of the average temperature ~ ( F oin) Fig. 22 shows that the development is much faster if both walls are cooled instead of one wall being nearly adiabatical (Bi, = - 1). The thermal development depends on the capacitance of the walls. In an example (Fig. 23) the outer wall of a circular Couette system is taken to be isothermal (9(1, Fo) = 0); the boundary condition at the inner wall is described by Bi, = - 1, ss,,= 0, and different values of the capacitance parameter Ci. The thermal development is delayed more, the larger the capacitance of the wall is taken to be. (K %

2 c , =o >,=O

, Bia=o

C,=IO

--

lo-’ Fourier number Fo

1

10

FIG.23. Thermal development of circular Couette flow depending on the capacitance parameter Ci at the inner wall; the outer wall is taken to be at constant temperature. Na = 1 ; m = 2; &.i = 0; S,,a = 0; Bii = - 1.

256

HORSTH. WINTER

FIG.24. Developing temperature in circular Couette flow for Ci = 0 and Ci = 0.1. The temperature of the outer wall is taken to be constant and equal to the initial temperature. K = 0.S;Na = 1 ; m = 2;Bii = -1.

The development of the temperature near the wall is determined by the value of C. For the example in Fig. 24 with C i = 0.1, the guide point initially is very close to the boundary. As the inner wall heats up, the guide point moves away from the boundary until the temperature of both the fluid and the wall reaches its full development. Dissipation and conduction balance and the temperature gradient at the inner wall becomes independent of Ci. a. Unsteady Plane Shear Flow with Closed Stream Lines. Analytical studies that include the time dependence of the viscosity q(fo, T , t ) do not seem to be available. Several authors calculated the developing temperature field in plane Couette flow of fluids with a viscosity independent of time: Gruntfest [89] : Krekel[86]: Powell and Middleman [92] :

q ( T )= q(To)e-B'T-To), ~ ( j T) , = _sinh-'

AY

v]

=

'

-

(C;TJ7

const.,

Winter [SS]: Practical applications of their studies are the Couette rheometer [88, 89,921 and a shearing device for breaking up particles suspended in a fluid [86].

VISCOUSDISSIPATION IN FLOWING MOLTENPOLYMERS

257

For the following example the assumption about the fully developed stress at t = 0 will be lifted. A Couette system is kept at rest, and the stress in the system is zero. At time t = 0 a shear experiment with constant shear rate is started. The shear stress 7 , z ( t )is found experimentally (see for instance [113]) to be governed by a time-dependent viscosity that increases gradually, goes through a maximum, and approaches a constant value. If these viscosity data are available, they might be used in the numerical program. For demonstrational purposes, the viscosity curve is approximated by q($, T, t )

=

[ ~ ~ ~ / ~ ) ( l ' m ) ~ ' e - " ' T --T e-'l')(l o)](l

L

,

Y

+ cZe-'l'),

(2.74)

Eq. (2.47)

which qualitatively fits the measured curve shapes. The maximum viscosity is chosen to be three times the viscosity of steady shear flow; the time of the maximum is chosen to be Fo = 0.1, i.e., at about one-tenth of the thermal development time. The time-dependent viscosity contains an elastic contribution, which, however, is not specified unless one uses a complete rheological constitutive equation. In the calculation of the dissipated energy, the elastic part of the work of the stress is taken to be negligible compared to the viscous part. The stress growth curve as chosen in Eq. (2.74) is reproduced by the numerical program with Na = 0.001 (dashed lines in Fig. 25). If viscous dissipation is important (Na = 1, for instance) the stress reaches an earlier maximum at a lower value; the general shape of the curve is not changed through the I

.

'\

.oN'

\NO =0.001 1

dimensionless time Fo

I Fourier number 1

FIG.25. Thermal influence calculated for the startup experiment of plane Couette flow with time dependent viscosity as described in Eq. (2.74). The walls are taken to be at constant temperature equal to the initial temperature; rn = 2.5.

HORSTH. WINTER

258

effect of dissipation, and rheological and thermal effects seem undistinguishable in stress growth experiments. For comparison, the developing shear stress curves for (rheologically) time-independent viscosity (as described in Eq. (2.47)) are calculated and drawn as solid lines in Fig. 25. b. Fully Developed Temperature Field. The fully developed case has drawn much attention (see Table V), which is due to a double-valued solution, found in 1940 by Nahme [44]for plane Couette flow of Newtonian fluids. The shear stress in fully developed circular Couette flow (including plane Couette flow as a limiting case) cannot exceed a certain value, even if the shear rate is very large; for shear stresses below the maximum possible value, there are always two feasible shear rates 9, a small one at high viscosity and low temperature and a large one at low viscosity and high temperatures. Changes from one shear rate to the corresponding one require large temperature changes, and due to the heat capacity of the system together with the small thermal conductivity of the polymer, oscillations between the two states do not seem possible. For demonstrating the double-valued solution, Nahme [44] used a dimensionless shear stress o* and a dimensionless shear rate $*, whose definition can be extended to power law fluids: t* =

Nal/(I+m)p RdR,

oo)/PR@(R,

j,* = Nam/(l+m)

O),

(2.75) (2.76)

PRe(R, 00) and PR,(R, 0) are the dimensionless shear stress (see Eq. (2.45))

of the fully developed temperature field and of the isothermal case, respectively; the ratio of the two is independent of R. The dimensionlessshear stress

dim1 schear stress T*

FIG.26. Shear rate f* as a function of shear stress 7* (both defined in Eqs. (2.76) and (2.75)) of the fully developed temperature field; the parameter is the geometry.

Viscous DISSIPATION IN FLOWING MOLTEN POLYMERS

259

x =0939 4..

x =05

go5 0)

c

0) rn

e

>

1

diml. shear stress

r*

FIG.27. Average temperature gm of the fully developed temperature field for different geometries of circular Couette flow. Both walls are at constant temperature (9, = 0); m = 2.

P,,(R, co) is a monotonically descreasing function of Na, and it cannot be used by itself to demonstrate the double-valued solution. As an example, in Figs. 26 and 27 the double-valued solution jJ*(z*) and the corresponding average temperature 9,(.r*) of the fully developed temperature field are shown for circular Couette flow at K = 0.5 and IC x 1. Each shear rate p* has only one corresponding temperature 9,.

4. Experimental Studies The gap width of Couette systems is fairly small; and it is very difficult, if not impossible, to measure the temperature distribution by conventional means. The wall temperatures, however, can be measured quite accurately; other quantities measured are the torque on the system, the rotational speed of the cylinders, and the geometry. The double-valuedness of the shear rate seems to have been verified by Sukanek and Laurence [55] only. For viscosity measurements, the shear rate is prescribed and the average velocity in plane Couette flow is taken to be ti = jh/2. The experiment should be performed at conditions close to isothermal, which means that the Nahme number should be as small as possible: (2.77) The Nahme number is proportional to the square of the gap width, i.e., the Couette system should have a very narrow gap. Manrique and Porter [57] built a Couette rheometer with a gap of 5 x mm; reportedly they could eliminate the influence of viscous dissipation up to shear rates of 3 x lo6 s- '.

HORSTH. WINTER 111. Elongational Flow; Shear Flow and Elongational Flow

Superimposed (Nonviscometric Flow)

The deformation during flow can be understood as a superposition of shear, elongation, and compression. If elongational components and density changes are negligible, the flow is shear flow, and the corresponding heat transfer problems can be analyzed as shown before. However, there are many engineering applications with a flow geometry different from shear flow; how the corresponding heat transfer problems are usually treated will be mentioned briefly. For a more detailed description, the reader will be referred to several examples in the literature. Other than for shear flow, there is no accepted rheological constitutive equation available for studying heat transfer. The proposed integral and differential constitutive equations are mostly tested in shear experiments at constant temperature, which might not be significant for nonviscometric flow during temperature changes. The main reason for not applying constitutive equations of elastic liquids is the fact that they require a detailed knowledge of the kinematics before the stress can be determined. But for other than Couette flow experiments, the kinematics of nonviscometric flows is not known in advance; it has to be calculated simultaneously with the stress. Presently a large emphasis of rheology is on solving nonviscometric flow problems at constant temperature. Rheological analysis is not advanced enough to incorporate temperature changes, and the present method of solution for nonviscometric engineering problems is practically identical with the one for steady shear flow, without care of the rheological differences. Elongational Flow Up to now, analytical studies on nonisothermal extensional flow have been done by means of a temperature dependent Newtonian viscosity, Eq. (1.13), and constant density. The studies are on melt spinning of fibers (see, for instance, [114,115]) and on film blowing (see for instance [116,1171). The measured stress and velocity indicate that the work of the stress a:Vv is very small (at least for film blowing [117]), and the heat transfer seems to be determined by convection with the moving film or thread and by conduction to the cooling medium. Shear Flow and Elongational Flow Superimposed In many different channel flows, as they occur in polymer processing, the rate of strain contains elongational components. The fluid elements are

VISCOUS

DISSIPATION IN FLOWING MOLTENPOLYMERS

261

FIG.28. Examples of converging and diverging flow: (a) Couette flow into a converging slit, which induces a pressure gradient for continuity reasons; (b) Couette flow in a converging annulus; (c) Poiseuille flow into a converging pipe or a converging slit; (d) radial flow in the gap between two parallel plates.

stretched while they are accelerated or slowed down along their paths. Examples (Fig. 28) are Couette flow into a converging slit or annulus, flow in a tapered tube, and radial flow between parallel plates. For describing the stress, one commonly uses the Strokes equation, Eq. (1.13), together with some average viscosity, or one takes the equation of the generalized Newtonian liquid, Eq. (1.14).The results of this kind of calculation seems to give relatively good estimates on temperature changes and viscous dissipation. Examples are heat transfer in screw extruders (see for instance [3, 102, 118-1221), in calendering [123], during mold filling [124-1291, and in melt solidification during flow [127-1291. If the deviations from shear flow are small, the stress might still be defined by the viscometric functions. An example of nearly viscometric flow is Poiseuilleflow in a pipe with constant but irregular cross section or Poiseuille flow in curved channels with constant cross section; the induced secondary flow in the cross section supports heat transfer toward the walls. The secondary flow, however, is very small. Whereas the improvement on the heat transfer for polymer solutions might be up to 30% [1303, for molten polymers (low density polyethylene in curved pipe) the influence of the secondary flow on the heat transfer was too small to be detectable with temperature probes in the melt [131]. Another example of nearly shear flow occurs in channels near a wall, even if the bulk of the fluid is mainly subjected to deformations other than shear

262

HORSTH.WINTER

[132]. For steady flow, the stress at the wall is described by the three viscometric functions and the wall shear rate, which of course can be determined only from the whole flow analysis including the nonviscometric part.

IV. Summary Heat transfer in flowing molten polymers is largely influenced by rheology, ie., by the rheological properties of the polymer and by the flow geometry. The rheology of steady shear flow is well understood, and hence the corresponding heat transfer problems can be treated most completely. However, heat transfer studies in flow geometries other than shear are, due to the present lack of an appropriate constitutive equation, only possible in very simplified form. The most important shear flow geometries are shown to be limiting cases of helical flow, and the corresponding heat transfer problems can be solved with one numerical program. Two groups of heat transfer problems are analyzed in the study: heat transfer in steady shear flow with open stream lines (represented by helical flow with a/& = 0) and the corresponding unsteady heat transfer problem with closed stream lines (represented by helical flow with d/dz = 0). The problem is completely determined by six dimensionsless parameters-the Nahme number; the Graetz number (or the Fourier number, respectively);the ratio of the radii of the annulus; the relative average axial velocity; the power law exponent of the viscosity; and the ratio of length to gap width-together with the boundary conditions. The commonly used: idealized boundary conditions are replaced by the Biot number for describing the heat conduction to the surroundings and by the capacity parameter for describing the thermal capacity of the wall during temperature changes with time. The conventional definition of the Nusselt number is not applicable to heat transfer problems with significant viscous dissipation, and a new definition has to be introduced. The shear dependence of the viscosity is described by a power law and the temperature dependence by an exponential function. The temperature coefficient of the power law region is shown to be directly related to the activation energy of the zero viscosity. ACKNOWLEDGMENT

The author thanks Prof. G. Schenkel for his critical advice and many helpful suggestions; he has supported not only this work but also several specific studies of the author which were incorporated here. The author thanks Profs. A. S. Lodge, E. R. 0.Eckert, and K. Stephan for

VISCOUS DISSIPATION IN FLOWING MOLTENPOLYMERS

263

many critical comments and the colleagues G. Ehrmann and M. H. Wagner for helpful discussions on details of the study. The Deutsche Forschungsgemeinschaft is also acknowledged for having enabled the author to spend the time from August 1973 to November 1974 in Madison at the Rheology Research Center which was a fruitful preparation for this work.

NOMENCLATURE a

Bi Cpr C"

C e

E

Fo

Gz

h k

1, L

l/h

m

M

Na Nu

P

P pRZ

4 r, R = r/r,

S thermal diffusivity [m2/s] Biot number [-I, see Eqs. r (2.21) and (2.26) T specific heat capacity at con- T stant pressure or at constant density [Jkg K] capacitance parameter of wall [-I, see Eqs. (2.26) and (2.30) internal energy [J/kg] activation energy [J/g-mole] D Fourier number, at/h2 [-I Graetz number, &h2/al [-] r, - ri = gap width [m]; h = r, for circular across sec- Y tion thermal conductivity [J/m s Z, Z U KI length of the slot power law exponent, see Eq. B (2.47) torque [mN] Nahme number, V 2 f i / k [-1, see Eq. (255) Nusselt number [-I, see Eq. s (2.63) pressure [N/m'], see Eq. E (1.13) dimensionless pressure gradient, see Eq. (2.43) 9 dimensionless shear stress components, see Eq. (2.44) and (2.45) specific heat flux at boundary [J/m2 s] radial coordinate (note: in Eqs. (1.9) and (2.12), R is the gas law constant) outer and inner radius of annulus [m]

wall thickness [m] time [s] temperature [K] average temperature [K], see Eq. (2.60) velocity components [m/s] angular velocity at inner wall [m/sl average velocity in z direo tion [m/s] reference velocity [m/s], see Eq. (2.38) dimensionless velocity components vep, v,P, vzp coordinate in r direction, see Eq. (2.32) = I/(/ GI) axial coordinate pressure coefficient of viscosity [m2/N1, 1- '(drt/aP)r.9 temperature coefficient of viscosity [K-'1, q-'(tlq/ 8T)P.V

rate of strain tensor [s-!] shear rate in simple shear flow [s-'1 unit tensor coefficient of thermal expansion, - p - '(dp/dT),

W-'l

dimensionless temperature, B(T - T o ) azimuth coordinate ratio of radii, rJra density [kg/m3] stress tensor [N/m2] extra stress tensor [N/m2] shear angle (see Fig. 1) first and second normal stress function in shear flow

HORSTH. WINTER INDICES 0

02

e

initial state, reference state, or related to the zero-viscosity (in a,, Po, E , ) fully developed state entrance

i, a r, R, z , Z , 0 S

W

inner or outer boundary coordinates surroundings wall, boundary of channel

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