Extrudate swell Reynolds number

For small Reynolds numbers (Re < 2), Newtonian fluids also show a slight extrudate swell and the measured values of Bex must be corrected accordingly [4]. 

A Three-Dimensional Finite-Element Analysis of the Effect of Reynolds Number on Extrudate Swell of Newtonian Liquids from Dies... 

Little has been published on the effect of Reynolds number on extrudate swell of liquids from fully three-dimensional dies. However, its effect on two-dimensional extrudate swell of a Newtonian fluid has been well characterized [1-4]. The influence of Reynolds number on the final die swell ratio for both planar jets (i.e., the thickness of the extrudate divided by the width of the channel) and axisymmetric jets (i.e., the diameter of the jet divided by that of the tube) are summarized in Fig. 1. In both cases, swell is greater than 1 for low Reynolds numbers but decreases to values less than 1 at high Reynolds numbers. As Reynolds number approaches infinity, the die swell ratios for the axisymmetric and planar jets approach the asymptotic values of /3/2 and 5/6, respectively [5,6]. 

Extrudate swell from more complicated, three-dimensional dies has been examined to some extent. However, most of these consider a zero Reynolds number situation [7-16]. The only fully three-dimensional die swell analyses including the effects of Reynolds which appear in the literature are those of Newtonian liquids from square dies [17-19]. One experimental study attempted to assess the effect of Reynolds number on extrudate swell from square and rectangular dies [20]. Unfortunately, those results appear to have been obscured by the inability to separate the effects of gravity from those of the Reynolds number [19]. 

Here, the three-dimensional finite-element procedure used by Gifford [19] to solve three-dimensional free-surface problems which include inertia will be reviewed. Previous results of simulations showing the effect of Reynolds number on extrudate swell of Newtonian liquids from a square die are discussed and compared to new results from a rectangular die. Limiting die swell solutions at infinite Reynolds number are discussed and the effect of the aspect ratio of the rectangle on these limiting values is presented. The effect of wall slip on die swell from the square die is also considered. 

As mentioned earlier, the effect of the Reynolds number on extrudate swell from square dies has been the subject of several computational investigations [17-19]. Those of Gifford [19] appeared to be the most detailed and are summarized here. 

Fig. 4 Extrudate swell from a square die as a function of Reynolds number.

Fig. 5 Maximum extrudate swell as a function of axial position and Reynolds number.

Figure 6 shows the effect of the Reynolds number on the final value of the maximum swell attained by the extrudate exiting the square dies. Gifford [19] showed that as the Reynolds number approaches infinity, die swell for the square case approaches an asymptotic value of 0.9611. Let us examine how such limits are obtained. 

In practice, few extrusion dies are designed with square channels at the exit. However, many are designed with a rectangular channel in the lip region exiting the die. The effect of Reynolds number on the extrudate swell from such shapes appears to have been heretofore unreported. 

Let us consider the case of extrusion from a channel with an aspect ratio of 4 shown schematically in Fig. 7. The same algorithm used to solve the above square-die swell problem is used. The final shape attained by the extrudate in this case is shown in Fig. 8 for several values of Reynolds number,... 

The effects of the Reynolds number on the extrusion of Newtonian fluid from square and rectangular dies has been considered. As with planar and axisymmetric jets, extrudates from three-dimensional dies swell at low Reynolds numbers but contract at high ones. Depending on its aspect ratio, limiting die swell from the rectangle varies that of the square (0.7255) and that of two-dimensional planar case (0.8333). Wall slip reduces die swell and in the cases of perfect slip, completely eliminates it. 

Gifford, W. A., A three dimensional finite element analysis of the effect of Reynolds number on extrudate swell of Newtonian liquids from square dies, Can. J. Chem. Eng. 77 161-167 (1993). 

Theory and experiment show that Newtonian liquids swell at low Reynolds numbers but shrink as inertia becomes important = 1.13 for Re < 2 and B = 0.87 for Re > 100 (Middleman, 1977). Typically elastic liquids are extruded in the low Reynolds range, so the Newtonian result is subtracted from the experimental values to give an elastic swell ... 

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