Introduction

Dealy [26] has described the capillary viscometer (or rheometer) is an instrument for measuring the flow properties of liquids. The common type of capillary rheometer for polymer melts consists of a temperature controlled barrel at the bottom of which is fitted a small die containing a capillary. There are two main types of capillary rheometer. The first is an instrument in which a plunger is forced down into the barrel (filled with polymer) at a constant rate and the pressure required is measured by a transducer or other appropriate means. Alternatively, an instrument may be used in which a load is applied to the top of the melt and the output rate for this load is measured. The load may be applied by means of weights or by the pressure of an inert gas [26].

Derivation of the shear strain rate and shear stress calculations used for capillary flow are covered in detail within many excellent texts [25, 26, 27, 28]. Dealy [26] has shown the apparent wall shear strain rate to be

The subscript w in equation 2.11 is used to denote that the shear rate is the shear rate at the wall. The subscript a in equation 2.11 is used to denote apparent shear rate because the derivation of equation 2.11 is based on an assumption that the velocity profile perpendicular to the flow direction is parabolic which is only true for Newtonian fluids [25]. A more general expression of the shear rate at the wall, expressed as the “true shear rate” at the wall has been derived by Rabinowitsch as follows [25]

1

Dealy [26] has shown the apparent wall shear stress to be AP r

V - " 2 f

(2'J3)

Brydson [25] has reported that these equations may only be valid if the following assumptions are made.

(i) That the flow pattern is constant along the full length of the capillary. (ii) That there is no slip at the capillary wall

(iii) That the fluid is time independent (iv) That the flow is isothermal (v) That the melt is incompressible

The validity of these assumptions, as well as a number of unusual flow phenomena noted by previous workers who have studied capillary flow, are reviewed in the sections below. 2.5.2 Die entrance effects and end corrections

The first of Brydson’s [25] assumptions for capillary flow, listed above, is that the flow pattern is constant along the full length of the capillary. During capillary extrusion of polymer melts a pressure drop in the die entrance region has been reported by a number of authors [25,26,27,28]. Similarly, a pressure drop at the die exit is also described within the literature [25,26,27,28].

Dealy [26] refers to the excess pressure drop at the die entrance as the “entrance pressure drop”, and the length of die required before fully developed flow commences as the “entrance length”. The entrance pressure drop generally makes a substantial contribution to the measured pressure [33]. It must therefore be allowed for in the analysis of the

extrusion rheometer results [26]. Isayev and Chung [34] have found these entrance and exit pressure drops to be a particular problem when studying the flow of polymer melts in short capillaries where a significant proportion of the total pressure drop may be due to these entrance effects. Bagley [35] has studied die entrance pressure drops and described a suitable method for correcting rheometer data to allow for such pressure drops. Bagley [35] demonstrated how a plot of pressure drop against die length to internal radius ratio at a fixed wall shear rate gives a straight line relationship. The intercept of these plots were described by Bagley [35] to be equivalent to the pressure drop due to die entrance and exit effects. The Bagley [29] plot gives the value of an end correction, e [26], where e is the die length to radius ratio as read from the Bagley [35] plot between the x-axis intercept and zero die length to radius ratio. Dealy [26] has modified the wall shear stress equation to take into account this end correction parameter e as follows

AP,

T . - ,. _{W 2 (l/r + e)}» ' (2.14)

Cogswell [27] has reported that a Bagley [35] end correction calculation requires extensive experimentation using dies of several different lengths. However, Cogswell [27] has suggested that in practice two dies may be adequate to determine end correction factors. When using two dies only for end correction Cogswell [27] recommends the use of a long die of length to internal radius ratio of 32 and an orifice restriction die of nominally zero length. The entrance and exit pressure drops are calculated by subtracting the pressure drop obtained using the zero die from that obtained using the long die. The shear stress at the wall allowing for die entrance and exit pressure drops by the zero die technique has been given by Cogswell [27] as

CAP, - AP0) r

The entrance pressure drop has been studied in considerable detail by La Mantia et al [36]. They [36] were able to derive entrance corrections as functions of temperature, weight-average molecular weight and molecular weight distribution. Bagley [37] has investigated the relationship between these “end corrections” and the shear stress. For polyethylene it was found that the relationship was linear, up to a critical shear stress at which extrudate distortion began.