Shear Rate at Wall

Since u = 0 when r = 2 , the first term in Eqnation 8.22 is zero. As a result, 

As discussed in the section 8.5.1, the shear stress is also related to r, which is the distance from the pipe center. From Equation 8.19, the following equation can be obtained  

Substituting Equation 8.24 and y = -dujdr into Equation 8.23 leads to  

By differentiating both sides with respect to r, the following equation can be obtained  

Substituting = RAPI2L into Equation 8.27 gives the Rabinowitsch equation  

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L = Die land length, mm P = Pressure drop across the die, MPa q = Volumetric flow rate, mmVsec Q = Extrusion rate, kg/hour g = Estimated shear rate at wall, sec ... 

For Newtonian flow (n = 1), the term 4g/s-R is in fact the true shear rate at all. However, for non-Newtonian flow, the term AQIn is known as the apparent shear rate at wall (y ), i.e.,... 

What are the shear stress and shear rate at wall when a polymer flows through a small-diameter pipe. 

Capillary viscometers are useful for measuring precise viscosities of a large number of fluids, ranging from dilute polymer solutions to polymer melts. Shear rates vary widely and depend on the instmments and the Hquid being studied. The shear rate at the capillary wall for a Newtonian fluid may be calculated from equation 18, where Q is the volumetric flow rate and r the radius of the capillary the shear stress at the wall is = r Ap/2L. 

Polymer melts are frequendy non-Newtonian. In this case the earlier expression given for the shear rate at the capillary wall does not hold. A correction factor (3n + 1)/4n, called the Rabinowitsch correction, must be appHed in such a way that equation 21 appHes, where 7 is the tme shear rate at the wall and nis 2l power law factor (eq. 22) determined from the slope of a log—log plot of the tme shear stress at the wad, T, vs 7. For a Newtonian hquid, n = 1. A tme apparent viscosity, Tj, can be calculated from equation 23. 

In most rotational viscometers the rate of shear varies with the distance from a wall or the axis of rotation. However, in a cone—plate viscometer the rate of shear across the conical gap is essentially constant because the linear velocity and the gap between the cone and the plate both increase with increasing distance from the axis. No tedious correction calculations are required for non-Newtonian fluids. The relevant equations for viscosity, shear stress, and shear rate at small angles a of Newtonian fluids are equations 29, 30, and 31, respectively, where M is the torque, R the radius of the cone, v the linear velocity, and rthe distance from the axis. 

For steady-state laminar flow of any time-independent viscous fluid, at average velocity V in a pipe of diameter D, the Rabinowitsch-Mooney relations give a general relationship for the shear rate at the pipe wall. 

The shear rate at the wall is obtained by letting r — R and the equation is more convenient if Vq is replaced by Q from (5.33). In this case... 

The correction factor for converting apparent shear rates at the wall of a circular cylindrical capillary to true shear rates is (3n + l)/4n, where n is the power law index of the polymer melt being extruded. 

Derive a similar expression for correcting apparent shear rates at the walls of a die whose cross-section is in the form of a very long narrow slit. 

Figure 9 Shear viscosity as a function of shear rate at the wall at 200°C, (O) Resin A, ( ) Resin B, (O) Resin C, (A) Resin D, (A) Resin E. (Refer to Table 2 for symbols code.) Source Ref. 56.

The power of this technique is two-fold. Firstly, the viscosity can be measured over a wide range of shear rates. At the tube center, symmetry considerations require that the velocity gradient be zero and hence the shear rate. The shear rate increases as r increases until a maximum is reached at the tube wall. On a theoretical basis alone, the viscosity variation with shear rate can be determined from very low shear rates, theoretically zero, to a maximum shear rate at the wall, yw. The corresponding variation in the viscosity was described above for the power-law model, where it was shown that over the tube radius, the viscosity can vary by several orders of magnitude. The wall shear rate can be found using the Weissen-berg-Rabinowitsch equation ... 

Viscosity measurements were carried out using Cannon-Fenske viscometers of appropriate capillary diameter so as to keep the efflux time between 200-300 seconds. Approximate shear rate at the wall was calculated using the equation... 

Figure 6. Viscosity as a function of concentration for PTF and BTF. Solvent n-hexane temperature 25°C. Number near each point indicates approximate shear rate at the wall.

The corresponding shear rate at the tube wall (yw) is given by... 

T shear rate at tube wall for Newtonian fluid, Eq. (3-16), [l/t]... 

This can be solved for the shear rate at the tube wall (yw) by first differentiating Eq. (6-92) with respect to the parameter rw by application of Leibnitz rule to give... 

The Hagen-Poiseuille equation [Eq. (6-11)] describes the laminar flow of a Newtonian fluid in a tube. Since a Newtonian fluid is defined by the relation r = fiy, rearrange the Hagen-Poiseuille equation to show that the shear rate at the tube wall for a Newtonian fluid is given by yw = 4Q/nR3 = 8 V/D. 

It is anticipated that the equilibrium filter cake mass would depend strongly on the axial velocity through the cross-flow filter assembly. The shear rate at the filter surface will increase the entrainment of the catalyst solids for a given permeate flow rate. Therefore, for each differential pressure condition, the axial velocity will be varied in order to quantify the effect of the wall shear on the filter cake resistance term. 

The second subscript N is a reminder that this is the wall shear rate for a Newtonian fluid. The quantity (8u/d,), or the equivalent form in equation 3.13, is known as the flow characteristic. It is a quantity that can be calculated for the flow of any fluid in a pipe or tube but it is only in the case of a Newtonian fluid in laminar flow that it is equal to the magnitude of the shear rate at the wall. 

Owing to the different relationship between t and y for a non-Newtonian fluid, the shear rate at the wall is not given by equation 3.13 but can be expressed as the flow characteristic multiplied by a correction factor as shown in Section 3.2. 

In order to determine the true shear rate at the wall it is necessary to use the Rabinowitsch-Mooney equation ... 

Another definition is based, not on the true shear rate at the wall, but on the flow characteristic. This quantity, which may be called the apparent viscosity for pipe flow, is given by... 

The minus sign has been placed inside the parentheses recognizing the fact that the shear rate y (equal to dvjdr) is negative. yw is the true shear rate at the wall and is related to the flow characteristic (8a/d,) by the Rabino-witsch-Mooney equation ... 

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