Study of mixing efficiency in roll-mills.

INTRODUCTION

The two roll-mill geometry, invented in 1835 by Edwin Chaffee for mixing additives into rubber as well as for manufacturing films and sheets, operates by processing a molten polymer between the rotating rolls (cylinders). The very simple and versatile design of the roll-mills [ILLUSTRATION FOR FIGURE 1 OMITTED] accounts for their wide use in compounding and calendering. The main difference between the roll-mills and the calenders is the ratio between the roll radius and the minimum gap separation between the rolls, which is much higher for calenders (200 to 500) than for the roll-mills (30 to 60).

Most of the hydrodynamic analyses for the two roll-mill flow geometry are extensions of Gaskell's (1) work using the lubrication approximation (2-6). More recent developments use numerical methods of calculation, particularly the finite element method (7-10). Manas-Zloczower et al. (11) attempted to model the dispersive mixing process in roll-mills. These authors followed the Takserman-Krozer (12) calculation of the flow field using bipolar coordinates and their own model of agglomerate rupture in simple shear flow to predict the dynamics of agglomerate size distribution.

In this paper, we used a fluid dynamics analysis package - FIDAP - based on the finite element method (13) to calculate the flow patterns in a two roll-mill geometry including the bank region. Dispersive mixing efficiency was analyzed in terms of shear stress distributions and elongational flow components. A frame invariant parameter was proposed to quantify the elongational flow components.

DESCRIPTION OF METHOD

Experimental observations show that the roll-mills lack lateral (axial) motion almost completely. Therefore a 2-D analysis will be sufficient to characterize the flow patterns. The field equations (Eqs 1 and 2) were solved for the isothermal flow of a power-law model fluid (Eq 3):

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

[Mathematical Expression Omitted]

where the power law index, n = 0.22, and the consistency index, m = 9.87 x [10.sup.4] N [multiplied by] [s.sup.0.22]/[m.sup.2], are describing the rheological behavior of a rubber compound with 65 parts SBR and 35 parts carbon black.

We used nonslip boundary conditions on the two roll surfaces (namely a tangential velocity [U.sub.1] on the upper roll surface and a tangential velocity [U.sub.2] on the bottom roll surface). In the bank region, the free surface was calculated from the condition of no normal flow across this surface, i.e.

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the unit vector normal to the free surface. For polymer melts of high viscosity and high capillary numbers, the surface tension does not play a significant role in the determination of the free surface (14).

The point of detachment of the polymer from the bottom roll and the corresponding blanket thickness on the upper roll were predicted based on Vlachopoulos's isothermal model for the calendering of power-law fluids (15). For a power index of 0.22, this model predicts a ratio [h.sub.1]/[h.sub.0] of 1.27.

The free surface was determined by first selecting a free surface configuration for a given amount of material and solving the flow problem while keeping the nodes on the free surface fixed in space. With the results obtained for the velocity field, a residue for the boundary condition on the free surface is calculated, and the free surface is modified accordingly. In general, three to five iterations are sufficient to calculate the free surface. Figure 2 presents the results obtained in four successive iterations (after the fourth iteration the global relative error in the velocity field was below 3%). Figure 3 shows the mesh design for the entire flow domain with 1832 quadrilateral elements with nine nodal points in each element for a total of 6661 nodal points.

In order to analyze the dispersive mixing efficiency of the flow field, we looked at the elongational flow characteristics and the distribution of shear stresses generated. There are numerous studies reported in the literature that point to the increased efficiency of elongational flows in blending immiscible liquids (16-20) or breaking solid agglomerates into a continuous matrix (21,22). Also the magnitude of shear stresses generated is a key factor to be considered in analyzing dispersive mixing efficiency.

In order to quantify the elongational flow components, we propose a frame invariant flow strength parameter, [S.sub.f], defined as

[Mathematical Expression Omitted]

where [Mathematical Expression Omitted] is the rate of deformation tensor and [Mathematical Expression Omitted] is the Jaumann time derivative of [Mathematical Expression Omitted], (e.g. the time derivative of [Mathematical Expression Omitted] with respect to a frame that rotates with the angular velocity of the fluid element). The flow strength parameter [S.sub.f] has been used in the past to characterize various types of steady flows (23). Its value ranges from 0 for pure rotational flow to infinite for pure elongational flow. For simple shear flow, [S.sub.f] has a value of 1.

The frame invariant flow strength parameter can also be used in constitutive equations describing the rheological behavior of fluids in various flow fields. One example is Larson's model (24), which differentiates between the polymeric fluid response in various strength flow fields:

[Mathematical Expression Omitted]

where the coefficients [[Alpha].sub.1] and [[Alpha].sub.2] depend on the flow strength [S.sub.f], m is a power law index, and [Mathematical Expression Omitted] is the magnitude of [Mathematical Expression Omitted].

For graphical purposes, we normalized the flow strength parameter [S.sub.f] as in Eq 7 below:

[N.sub.s] [equivalent to] [S.sub.f]/1 + [S.sub.f] (7)

where [N.sub.s] ranges from 0 to 1.

RESULTS AND DISCUSSION

Velocity profiles in the nip region between the two rolls and in the bank region are shown in Fig. 4. The boundary conditions used were a rotational speed of 40 cm/s for both rolls and a volume of material on the rolls equal to [1.609V.sub.min], where [V.sub.min] = 2[Pi][Rh.sub.0] is the minimum amount (per unit width) of polymer obtained considering a blanket thickness equal to the minimum gap separation. The development of a vortex in the bank region can be observed. The pressure contours for the same boundary conditions are shown in Fig. 5. The maximum pressure is generated in the nip region at x = -1.05 cm.

We have found that for an amount of material on the rolls exceeding approximately 1.5[V.sub.min], one or more vortices are developed in the bank region. Figures 6 and 7 compare the streamline contours for V = 1.194[V.sub.min] [ILLUSTRATION FOR FIGURE 6 OMITTED] and V = 1.609[V.sub.min] [ILLUSTRATION FOR FIGURE 7 OMITTED]. Later on we will show that the presence of vortices in the bank region has no beneficial effect on dispersive mixing efficiency.

Figures 8 and 9 show the distribution plots for the parameter [N.sub.s] and shear stress in the bank and nip regions of the roll-mill (for conditions similar to [ILLUSTRATION FOR FIGURE 4 OMITTED]). A strong (elongational) flow (high values of [N.sub.s]) is characteristic for the converging area to the nip region. This region shows also high shear stresses. In fact, it is the Y-shaped region clearly pictured in Fig. 9 that exhibits good flow characteristics for dispersive mixing (high values for [N.sub.s] and shear stress). The nip region shows overall low mixing capability in spite of the two high shear stress stripes near the roll surfaces. Also, the vortex region, exhibiting more rotational flow and low shear stresses is not effective in mixing. Figures 10 and 11 show the volumetric distributions for the parameter [N.sub.s] and shear stress. These Figures indicate a broad distribution of flow regimes for the roll-mills.

We looked at the influence of the friction ratio (ratio of the tangential velocities [U.sub.1]/[U.sub.2] for the two rolls) on the mixing efficiency of the roll-mills. Table 1 gives the average flow field characteristics for three different friction ratios (the average values were obtained by weighing the corresponding parameters for each element by the area of the element itself for the flow domain). The last two columns in the Table were calculated based on the concept of better mixing. In the mixing process, we would like the material to experience both high shear stresses and elongational flow at the same time. Although increasing the friction ratio generates higher shear stresses, the overall mixing performance, as judged from the last two columns in Table 1, is not improved.

SUMMARY AND CONCLUSIONS

In this paper, we presented the results of 2-D, isothermal flow simulations of a power-law model fluid in a two roll-mill geometry with a free surface in the bank region. The flow patterns were obtained by using a fluid dynamics analysis package based on the finite element method. The flow field was characterized for dispersive mixing efficiency in terms of shear stresses generated and a parameter quantifying elongational flow components. This last parameter was defined in a frame invariant manner, which makes it useful for further use in constitutive equations describing the rheological behavior of fluids as a function of the flow [TABULAR DATA FOR TABLE 1 OMITTED] field strength. We found that the converging region rather than the nip region provides better mixing flow characteristics. We also found that the presence of vortices in the bank region is not beneficial for dispersive mixing. The overall mixing performance is not improved by increasing the friction ratio between the two rolls.

ACKNOWLEDGMENT

The authors would like to acknowledge the use of computing services of the Ohio Supercomputer Center.

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