Shear stress INDEX

Suppose we divide the flow segments into classes according to relaxation times and index the various states by the subscript i. Thus the relaxation time and the component of shear stress borne by the segments in class i are and Fj, respectively. The applied shear force is related to the Fj s through... 

A slit die is designed on the assumption that the material is Newtonian, using apparent viscous properties derived from capillary rheometer measurements, at a particular wall shear stress, to calculate the volumetric flow rate through the slit for the same wall shear stress. Using the correction factors already derived, obtain an expression for the error involved in this procedure due to the melt being non-Newtonian. Also obtain an expression for the error in pressure drop calculated on the same basis. What is the magnitude of the error in each case for a typical power law index n = 0.377... 

If the velocity profile is the same for all stream velocities, the shear stress must be defined by specifying the velocity ux at any distance y from the surface. The boundary layer thickness, determined by the velocity profile, is then no longer an independent variable so that the index of < in equation 11.25 must be zero or ... 

Hydraulic fracturing fluids are solutions of high-molecular-weight polymers whose rheological behavior is non-Newtonian. To describe the flow behavior of these fluids, it is customary to characterize the fluid by the Power Law parameters of Consistency Index (K) and Behavior Index (n). These parameters are obtained experimentally by subjecting the fluid to a series of different shear rates (y) and measuring the resultant shear stresses (t). The slope and Intercept of a log shear rate vs log shear stress plot yield the Behavior Index (n) and Consistency Index (Kv), respectively. Consistency Indices are corrected for the coaxial cylinder viscometers by ... 

The latter form is required to reflect the fact that the direction of the shear stress must reverse when the shear rate is reversed, and to overcome objections such as y , and therefore r, having imaginary values when y is negative. The power n is known as the power law index or flow behaviour index, and K as the consistency coefficient. 

Figure 3.19 Calculation of power law index n from the shear stress at the wall (tJ as a function of the apparent shear rate ()...

The melt Index test measures the flow property at a fixed wall shear stress In the capillary. The shear stress depends on the load specified for the condition and it is provided in Table 3.8. The apparent shear rate at the capillary wall that the resin experiences depends on the Ml value measured, and it can be calculated using a modification of Eq. 3.33 as follows ... 

Table 3.8 Test Conditions and Shear Stress at the Capillary Wall for Melt Index Testing as Prescribed by ASTM D1238-04c [15]...

Two main types of viscometers are suitable for the determination of the viscosity of a polymer melt The rotation viscometer (Couette viscometer, cone-plate viscometer) and the capillary viscometer or capillary extrusiometer. The latter are especially suitable for laboratory use since they are relatively easy to handle and are also applicable in the case of high shear rates. With the capillary extrusiometer the measure of fluidity is not expressed in terms of the melt viscosity q but as the amount of material extruded in a given time (10 min). The amount of ex-trudate per unit of time is called the melt index or melt flow index i (MFI). It is also necessary to specify the temperature and the shearing stress or load. Thus MFI/2 (190 °C)=9.2 g/10 min means that at 190 °C and 2 kg load, 9.2 g of poly-... 

The melt flow index is a useful indication of the molar mass, since it is a reciprocal measure of the melt viscosity p. p depends very strongly on 77 ( ) (doubling of results in a 10.6 times higher 77 ). This relation is valid for the zero-shear viscosity the melt index is measured at a shear stress where the non-Newtonian behaviour, and thus the width of the molar mass distribution, is already playing a part (see MT 5.3.2). The melt index is a functional measure for the molar mass, because for a producer of end products the processability is often of primary importance. 

This index describes the potential of the tablet to cap and laminate, and is related to the ability to relieve shear stresses within the compact via material flow. In Equation (1), TS equals the tensile strength of the normal tablet and TSo is measured from a tablet with a hole in its center, which simulates a defect. 

Some fermentation broths are non-Newtonian due to the presence of microbial mycelia or fermentation products, such as polysaccharides. In some cases, a small amount of water-soluble polymer may be added to the broth to reduce stirrer power requirements, or to protect the microbes against excessive shear forces. These additives may develop non-Newtonian viscosity or even viscoelasticity of the broth, which in turn will affect the aeration characteristics of the fermentor. Viscoelastic liquids exhibit elasticity superimposed on viscosity. The elastic constant, an index of elasticity, is defined as the ratio of stress (Pa) to strain (—), while viscosity is shear stress divided by shear rate (Equation 2.4). The relaxation time (s) is viscosity (Pa s) divided by the elastic constant (Pa). 

The shear-stress convention is a bit more complicated to explain. In a differential control volume, the shear stresses act as a couple that produces a torque on the volume. The sign of the torques defines the positive directions of the shear stresses. Assume a right-handed coordinate system, here defined by (z, r, 9). The shear-stress sign convention is related to ordering of the coordinate indexes as follows a positive shear xzr produces a torque in the direction, a positive xrg produces a torque in the z direction, and a positive x z produces a torque in the r direction. Note also, for example, that a positive xrz produces a torque in the negative 6 direction. 

Index Entries Com stover rheological measurement shear stress shear rate non-Newtonian fluids Power Law parameters. 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

First, the rate of shear, which is not linear with the shearing stress due to the non-Newtonian behaviour, varies with the different types of polymer. The processability of different polymers with an equal value of the MI may therefore differ widely. An illustration of this behaviour is given in Fig. 15.14. Furthermore the standard temperature (190 °C) was chosen for polyethylenes for other thermoplastics it is often less suitable. Finally, the deformation of the polymer melt under the given stress is also dependent on time, and in the measurements of the melt index no corrections are allowed for entrance and exit abnormalities in the flow behaviour. The corrections would be expected to vary for polymers of different flow characteristics. The length-diameter ratio of the melt indexer is too small to obtain a uniform flow pattern. 

The rheological properties of pastes of emulsion PVC having k-values of 69,70 and 75 for coating fabrics were investigated using a coaxial viscometer and the influence of the content and type of plasticiser (dioctyl phthalate and dioctyl adipate) on these properties evaluated. Consistency index and power law index values for the various formulations were determined and the results obtained analysed statistically using shear stress as the variable for each paste. 14 refs. 

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