## Eq

where Do and D are the initial and final diameters, respectively.

Fig. 8 Circular square entry die. Source: Ref 6

The flow through the die land is resisted by drag forces along the die wall, that is, friction force between the feedstock and the die wall. The drag force acts over the perimeter of the die land. The pressure drop (AP2) required to cause flow through the die land can be derived from a simple force balance (Ref 6):

AP2A = TfML

where Tf is the extruded part-wall shear stress (assumed to be independent of the position on the perimeter), and A. M, and L are the cross-sectional area, perimeter, and length of the die land, respectively. For flow through a die with a circular cross section, a geometry used for wire and bar production, the pressure drop is given as:

where D is the diameter of the die land. For flow through a die land with a square cross section, a geometry used to make brick and tile, the following equation represents the pressure drop:

where b and c are the depth and width of the rectangular cross section. Finally, for flow in the space between two annular cylinders, a geometry used to make tubular products, the equation for the pressure drop is given by:

where Dx - D2 are the outer and inner diameters of the tube, respectively.

The total pressure drop (A.P) required for feedstock flow through a circular, square entry die is (Ref 6, 25):

where iJ b and Tf are related to the flow characteristics of the feedstock. Experimentally (Ref 25), it has been found that iJ

b = (Ga + Q v") and Tf = (T0 + which is very similar in form to Eq 1. Here, <T0 is the yield stress, extrapolated to zero velocity; £Vcharacterizes the effect of velocity on the yield stress; Tf is the initial wall shear stress; 1 "'is a factor accounting for the velocity dependence of wall shear stress; n and m are shear-thinning exponents; and v is the feedstock velocity. The values of feedstock parameters (ff0, T0, ft, \$. m, and n) are determined from ram extrusion experiments in which extrusion pressures are measured through two dies of differing die-land lengths and at a range of extrusion velocities. For a circular square entry die the extrusion pressure can be calculated as:

The pressure calculated using Eq 7 is compared to the measured pressure for an alumina extrusion in Table 4. Differences in calculated and measured pressures may be attributed to the die-entry velocity being a function of the shear rate factor and shear-thinning exponent (Ref 27).

 Extrusion velocity, Extrusion pressure, kPA mm/s Measured Calculated using Eq 7 Square entry—circular die land (Do = 25.4 mm; D = 9.5 mm; L = 50.8 mm) 1.3 5.05 5.0 2.7 6.26 5.6 5.3 7.74 6.5 13.3 11.00 9.6 Square entry—square die land (Ao= 506 mm2; A = 9.0 mm2; b = 3.0 mm; c = 3.0 mm; L = 3 mm) 4.7 1.89 1.23 18.7 3.58 2.27 46.7 5.21 4.53

Note: Feedstock: 83 wt% A1203, 3.5 wt% clay, 6 wt% water, 7.5 wt% glucose. <J0 = 24 kPa, G= 18 kPa • s/mm;

Note: Feedstock: 83 wt% A1203, 3.5 wt% clay, 6 wt% water, 7.5 wt% glucose. <J0 = 24 kPa, G= 18 kPa • s/mm;

0=18

Conical or tapered dies are frequently used in extrusion processes. For flow through a conical die entry, the feedstock is subjected to both compressive and shear stresses, with the generation of significant drag forces along the die wall. The pressure drop (APi) for flow through a die with a conical entry can be written as (Ref 6):

where £?is the die entrance angle (note that for a square entry die 0= 90° and the cot 0= 0; thus Eq 8 reverts to Eq 4). The flow velocity in the tapered die is a function of both radial and axial location. The material nearest the central axis experiences more acceleration, and experiments utilizing striated feeds of different colors (Ref 30), and monitoring extrusion with nuclear magnetic resonance (Ref 29), indicate that the differential flow velocity between the center and wall of the die increases with increasing die entrance angle (as shown in Fig. 9).

Fig. 9 Influence of die taper on the extrudate velocity profile. Source: Ref 30

Analysis of the preceding equations reveal that the extrusion pressure will be high when the extrusion velocity is high and when the ratio of the die land length to die diameter (LID) is large, which is consistent with practice (Fig. 10). The extrusion pressure will also be high when the reduction ratio (D/D0) is large and when the die entry angle (0) is small (Fig. 11). Finally, the extrusion pressure will be high when the friction in the die land is high (Fig. 12). From Fig. 12, the addition of lubricants decreases the frictional force along the die land, and thus reduces the work required for feedstock flow.

0 LL

 - L/O-fl L/D =1 —- .*—■ 1 ' 1 i

10 20 30 40 50 Eitrudate velocity ^(mm s-1}

Fig. 10 Influence of extrudate velocity and die land aspect ratio (L/D) on extrusion pressure. Extrusion of AI2O3 powder; binder is clay, starch, and water; Do = 25.4 mm, D = 3.2 mm. Source: Ref 6

Fig. 11 Influence of die semiangle and reduction ratio on extrusion pressure (extrusion of ZrO2 powder; binder is hydroxypropyl methylcellulose and water). Source: Ref 28

Fig. 12 Influence of lubrication on extrusion pressure. Source: Ref 1

A more generalized form of Eq 7 and 8 is given by Eq 9 (Ref 6, 26):

0 0