True Shear Stress

We have presented the important corrections needed to get true shear rate from flow rate measurements. We now turn our attention to getting the ccurect shear stress. According to the assumptions used in deriving eq. 6.2.3, we need to measure Pc/L, the pressure 

Schematic of pressure profile in a capillary rheometer. Where, 

Pf pressure drop due to friction between piston reservoir walls 

pressure drop in the reservoir, due to steady, fully developed flow Pe previous drop in the capillary due to converging flow from reservoir to capillary Pc, pressure drop due to nearly fully developed flow in capillary 

Ptf non-zero pressure at capillary exit due to fluid elasticity 

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F(FG = normal (shear) component of force A = area u(w) = normal (shear) component of displacement o-(e ) = true tensile stress (nominal tensile strain) t(7) = true shear stress (true engineering shear strain) p(A) = external pressure (dilatation) v = Poisson s ratio = Young s modulus G = shear modulus K = bulk modulus. 

The calculations performed must be seen as no more than approximations, since they are based on the assumption of Newtonian behaviour, something from which polymer dispersions are far away. For this reason the instrument offers two correction methods the Bagley method for determining the true shear stress t and the Rabinowitsch method for correcting the shear rate. 

By plotting the extrusion pressure versus the length-to-diameter ratio, at constant shear rate, the so-called Bagley correction (23) can be determined, as shown in Figure 8. The Bagley correction e allows the true shear stress o to be calculated following - AP... 

In normal capillary rheometry for polymer melts, the flowing stream exits into the atmosphere, and the driving static pressure in the reservoir is taken to he AP. In such cases, end effects involving viscous and elastic deformations at the entrance and exit of the capillary should be taken into account when calculating the true shear stress at the capillary wall, particularly if the ratio of capillary length to radius L/R) is small. 

The following capillary viscometer data on a high pressure polyethylene melt at 190°C have been reported in the literature [A.P. Metzger and R.S. Brodkey, J. Appl. Polymer Sci., 1 (1963) 399], Obtain the true shear stress-shear rate data for this polymer. 

As can be seen, the value of the correction factor (in + l)/(4 ) varies from 25% to 55.6%. Thus, the values of (t, Yw) represent the true shear stress-shear rate data for this polymer melt which displays shear-thinning behaviour as can be seen from the values of < 1. 

The following volumetric flow rate - pressure gradient data have been obtained using a capillary viscometer (D = 10 mm and L = 0.5 m) for a viscous material. Obtain the true shear stress-shear rate data for this substance and suggest a suitable viscosity fluid model. 

Polymer melts are frequently non-Newtonian. In this case the earlier expression given for the shear rate at the capillary wall does not hold. A correction factor (3n -I- l)/4n, called the Rabinowitsch correction, must be applied in such a way that equation 21 applies, where j>tw is the true shear rate at the wall and n is a power law factor (eq. 22) determined from the slope of a log-log plot of the true shear stress at the wall, ttw, vs For a Newtonian liquid, n = 1. A true apparent viscosity, r]t, can be calculated from equation 23. 

As a result of these difficulties, typical capillary rheometers measure pressure in or above the reservoir as indicated in Figure 6.2.1 or from the forces on a driving piston. (Different capillary rheometer designs are discussed further in Chapter 8.) To determine the true shear stress, a number of corrections must be considered. 

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