Parallel Disk

Sample preparation and loading is simpler for very viscous materials and soft solids 

Can vary shear rate (and shear strain) independently by rotation rate Q (and 6 ) or by changing the gap h permits increased range with a given experimental set up 

Measure N2 when used with cone and plate thrust data 

Preferred geometry for viscous melts for small strain material functions 

As in wide gap Couette and Poiseuille flow (Chapter 6), shear rate is not constant. Thus we must use a derivative to relate shear stress to total torque. The resulting equations are given below and then derived in the remainder of this section. 

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When compared to standard (open cavity) cone-plate or parallel disks rheometers, closed cavity torsional rheometers such as the RPA or the PPA have unique high-strain capabilities, which prompted us to modify the instmment in order to investigate the promises of FT rheometry, as outlined a few years ago by the pioneering works of Wilhelm. The technique consists of capturing strain and torque signals and in using FT calculation algorithms to resolve it into their harmonic components, as detailed below. 

Yoshimura, A.S. Prud homme, R.K. "Viscosity Measurements in the Presence of Wall Slip in Capillary, Couette, and Parallel-Disk Geometries," SPE Reservoir Engineering, May 1988, 735-742. 

Consider the steady-state, fully developed, incompressible flow between parallel disks, such as illustrated in Fig. 5.9. In concert with the Jeffery-Hamel assumptions that were made in the previous configurations, one can assume that only the radial velocity is nonzero. As a consequence the continuity and momentum equations reduce to the following ... 

Fig. 5.9 Illustrative geometry for the radial flow between parallel disks.

Fig. 5.11 The T( 2) function for radially inward flow between parallel disks.

In Section 5.5 it was shown that for radial flow between parallel disks that the convective term could not be retained. It appears here that for 0 0 the convective term can be retained, since the cross-stream momentum equation provides a relationship between / and the pressure gradient. In the limiting case of 0 = n/2 the convective terms vanishes in any case. 

As a partial check on the derivations in the conical coordinates, it should be possible to recover two, easily identified, special cases—the radial flow between parallel disks and the axial Poiseuille flow in an cylindrical annular gap. The parallel-disk flow (Section 5.5) is the case where 0 = 0, with x taking the role of r and y taking the role of z. In this case, h = De/2 + x = r. The momentum equations become... 

U. Yilmazer and D.M. Kalyon, Slip Effects in Capillary and Parallel Disk Torsional Flows of Highly Filled Suspensions, J. Rheol., 33 1197-1212 (1989). 

A. Yoshimura and R.K. Prud homme, Wall Slip Corrections for Couette and Parallel Disk Viscometers, J. Rheol., 32 53-67 (1988). 

Let us consider a Newtonian fluid that is flowing due to a pressure gradient between two parallel disks that are separated by a distance 2h. The velocity and pressure fields that we will solve for are ur = ur z, r) and p = p(r). According to the Newtonian fluid model,... 

Torsional Drag Flow between Parallel Disks Solve the torsional drag flow problem between two parallel disks, one of which is stationary while the other is rotating with an angular velocity 0(s ). Note ve/r = constant.)... 

Radial Pressure Flow between Parallel Disks Solve the problem of radial pressure flow between two parallel disks. The flow is created by a pressure drop (P r 0 P r R). Disregard the entrance region, where the fluid enters from a small hole at the center of the top disk. 

Non-Newtonian Viscosity In the cone-and-plate and parallel-disk torsional flow rheometer shown in Fig. 3.1, parts la and 2a, the experimentally obtained torque, and thus the % 2 component of the shear stress, are related to the shear rate y = y12 as follows for Newtonian fluids T12 oc y, implying a constant viscosity, and in fact we know from Newton s law that T12 = —/ . For polymer melts, however, T12 oc yn, where n < 1, which implies a decreasing shear viscosity with increasing shear rate. Such materials are called pseudoplastic, or more descriptively, shear thinning Defining a non-Newtonian viscosity,2 t],... 

Torsional Flow of a CEF Fluid Two parallel disks rotate relative to each other, as shown in the following figure, (a) Show that the only nonvanishing velocity component is vg = flr(z/H), where ft is the angular velocity, (b) Derive the stress and rate of deformation tensor components and the primary and secondary normal difference functions, (c) Write the full CEF equation and the primary normal stress difference functions. 

The barrel surface exerts a retarding effect on flow rate, just as the flights in a screw extruder do. Edelist and Tadmor (45) derived the shape correction factors for Newtonian fluids, which are plotted for parallel disks as a function of H/ (Rd — Rs) in Fig. 6.32 and for wedge-shaped disks in Fig. 6.33. 

Example 6.14 Squeezing Flow between Two Parallel Disks This flow characterizes compression molding it is used in certain hydrodynamic lubricating systems and in rheological testing of asphalt, rubber, and other very viscous liquids.14 We solve the flow problem for a Power Law model fluid as suggested by Scott (48) and presented by Leider and Bird (49). We assume a quasi-steady-state slow flow15 and invoke the lubrication approximation. We use a cylindrical coordinate system placed at the center and midway between the plates as shown in Fig. E6.14a. 

Fig. E6.14b Dimensionless plot of squeezing flow data by Leider (50) representing 181 runs for four fluids silicone oil, 1% solution of hydroxyethyl cellulose (HEC), 0.5% solution of Separan (polyacrylamide) in glycerin, and polyisobuthylene solution. [Reprinted by permission from R J. Leider, Squeezing Flow between Parallel Disks, II, Experimental Results, Ind. Eng. Chem. Fundam., 13, 342-346 (1974).]...

P. J. Leider, Squeezing Flow between Parallel Disks II Experimental Results, Ind. Eng. Chem. Fundam., 13, 342-346 (1974). 

Fig. 9.49 The evolution of the interfacial area of a viscous Thiokol rubber in a 26.6-cm parallel-disk mixing chamber with a. — 0.5, with the number of turns. The rubber filled up half the chamber with one-quarter cream color (at the channel block at the left side) and one-quarter black. The numbers on the figure indicate the number of turns from 1/4 to 10. [Reprinted by permission from B. David and Z. Tadmor, Laminar Mixing in Co-rotating Disk Processors, Int. Polym. Process., 3, 38-47 (1988).]...

Example 11.1 Chain Modification (Branching and Partial Cross-linking) of PET with Triglycidyl Isocyannrate (TGIC) Dhavalkikar (39) conducted the reaction cited in the Example title on samples placed between the rheometrics mechanical spectrometer (RMS) parallel disks in the temperature-controlled chamber under nitrogen. He followed the reaction dynamics chemorheologically by measuring, in-line, the in- and out-of-phase dynamic moduli G (t) and G"(t) they are indicative of the elastic and viscous nature of the molten reactive samples. 

Fig. Ell.le The coupled diffusion-reaction process while applying a steady torsional parallel-disk flow of y — 1 s-1 to the initially segregated sample.

Somewhat similar is a continuous GC method that does not use a packed bed.7 It consists of a pair of parallel disks on which the stationary phase is coated. The sample enters at the center, and both disks are rotated. 

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