Equations Rabinowitsch

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 ... 

The calculation of the shear rate at the capillary wall, 7 , is computed from the function slope of Fig 3.18 and the apparent shear rate using Eq. 3.36. The derivative of the function appears relatively constant over the shear stress range for Fig. 3.18. Many resin systems will have derivatives that vary from point to point. The corrected viscosity can then be obtained by dividing the shear stress at the wall by the shear rate i ,. Equation 3.36 is known as the Weissenberg-Rabinowitsch equation [9]. 

EitherEq. E3.1-9or Eq. E3.1-10, known as the Rabinowitsch or Weissenberg-Rabinowitsch equations, can be used to determine the shear rate at the wail yw by measuring Q and AP or r and Tw (21). Thus, in Eq. E3.1-4 both xw and yw can be experimentally measured for any fluid having a shear rate-dependent viscosity as long as it does not slip at the capillary wall. Therefore, the viscosity function can be obtained. 

The Rabinowitsch equation has been used in the long capillary viscometry data found in Appendix A. Figure E3.1 shows long capillary tw vs. TH. and rw vs. yw results with and without the Rabinowitsch correction. 

The Rabinowitsch Equation for Fluids Exhibiting Slip at the Wall Derive the Rabinowitsch equation for the case where the fluid has a slip velocity at the wall Vw. [See L. L. Blyler, Jr., and A. C. Hart, Polym. Eng. ScL, 10, 183 (1970).]... 

In the case of a non-Newtonian fluid, the true wall shear rate Yww can be calculated by using a procedure similar to the Rabinowitsch equation for capillary flow, obtaining... 

This is the Rabinowitsch equation. Thus, determining Q and dQ/dAP as a function of AP, it is possible to determine y, the shear rate at the wall (eqn 7.19) the shear stress at the wall is also known (from eqn 7.13). It is then possible to plot wall shear stress versus wall shear rate and to determine the apparent viscosity. 

It is usual, in practice, to ignore the Rabinowitsch equation and to assume that the velocity profile for the polymer liquid is parabolic to within the required error. With this assumption the wall shear rate is taken to be given by the Newtonian value, so that, from eqns 7.13 and 7.17, the apparent viscosity is... 

For polymer solutions that are in the non-Newtonian flow region for the shear rate range of the capillary, the shear rate calculated with Eq. (3.4) has to be corrected. The true shear rate at the capillary wall can be determined using the Weis-senberg-Rabinowitsch equation [15] ... 

The combination of Equations 3.86 and 3.88 for and >/w(7w) e known as the Mooney-Weissenberg-Rabinowitsch equations. They show how a rj/y curve may be extracted from capillary flow data (i.e. pressure drop/flow rate measurements) for a non-Newtonian fluid in steady shear flow and these are the equations already presented without proof in Figure 3.14. 

However, there is a condition which must be fulfilled for the above analysis to be valid. It is that the shear rate at a given radius in the tube is a unique function radius. This will normally be so if the tube radius is large compared with the molecular dimension of the polymer. However, for very narrow capillaries, this may not be the case and the solution may become depleted in polymer molecules close to the capillary wall through the depleted layer effect (see Chapters 6 and 7). Thus, the concentration may vary across the capillary, and hence the constitutive model relating rj and y must also depend on local concentration and there is not a unique inversion of the rj/y relationship. This will be discussed in detail in Chapter 6, which will refer back to the development of the Mooney-Weissenberg-Rabinowitsch equations in this context (Sorbie, 1989, 1990). 

Alternatively, Tw can be measured directly by using a single long capillary with l/r about 40. The velocity gradient in Fig. 3-5 is assumed to be parabolic, but this is true strictly only for Newtonian fluids. The Rabinowitsch equation [24] corrects for this discrepancy in non-Newtonian flow, such as that of most polymer melts ... 

Parallel-plate torsional flow is a second choice. Assuming incompressible flow, the viscosity can be calculated from the total torque needed to turn one disk while keeping the other immobile. Following a derivation similar to that used for the Weissenberg-Rabinowitsch equation and using Leibnitz rule, it is straightforward to get the viscosity at the rim of the disk ... 

Substituting = RAPI2L into Equation 8.27 gives the Rabinowitsch equation ... 

The Rabinowitsch equation allows the calculation of the shear rate at the pipe wall from three measurable quantities R, Q, and AP. Since the flow properties of a fluid are independent of the pipe structure, the curve of the fluid is the same as the r-y curve. Therefore, by using Equation 8.18 and the Rabinowitsch equation, the flow curve, i.e, t-y or log r-log y, can be obtained. However, the use of the Rabinowitsch equation sometimes is tedious, and hence it often is simplified by using the following mathematical rearrangements. 

This simple equation is known as ibeMetzner form of the Rabinowitsch equation. Now, the flow curve (r-y or log r-log y ) of the fluid can be obtained by using Equations 8.18 and 8.32. 

The Metzner form of the Rabinowitsch equation (Equation 8.32) can then be rewritten to ... 

This equation is the Weissenberg-Rabinowitsch equation [10,11] it facilitates the measurement of the wall shear rate from Q versus Ap data for any fluid that satisfy the assumptions leading up to Equation 8.4. 

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