Parallel disk geometry

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. 

The parallel disk geometry, also called the parallel plate geometry, consists of two disks with radius ro separated by the gap h (Figure 3-8). Assuming steady, laminar, and isothermal flow, the expression for shear rate is ... 

Figure 3-8 Schematic Diagram of a Parallel Disk Geometry.

The parallel disks geometry is sketched in Figure 5.5.1. If we assume ... 

It is easier to load and unload viscous or soft solid samples with the parallel disk geometry than with cone and plate or concentric cylinders. Thus parallel disks are usually preferred for measuring viscoelastic material functions like G(r, y), G (), or 7(r, t) on polymer melts. To evaluate moduli or compliance, we use the strain and stress at the edge of the disk (eqs. 5.5.5 and 5.5.8), but now the stress must be corrected by d In A//d In y (Soskey and Winter, 1984). In the linear viscoelastic region G(r, y) = G(r) anddln Af/dlny = 1. 

Another use for data collected at different gaps in the parallel disk geometry is in determining wall slip. Yoshimura and Prud homme (1988) have shown that the difference in apparent stress versus shear rate at two different gaps can be related to wall slip, analogous to eq. 5.3.27. 

Accurate data can be obtained with the parallel disk geometry. Figure 5.5.2 shows some parallel disk total dmist data for Ai — N2 calculated by eq. 5.5.17. There is reasonable agreement between N2 by the difference between parallel disks and cone and plate and by birefringence studies. 

Pa in 90 seconds. The response time of the system with parallel disk geometry is... 

Liquid convection baths are best suited for the concentric cylinder geometry and for temperatures between —10 and 1S0°C. Temperature gradients with cone and plate and parallel disk geometries can be a problem because of heat loss from the upper fixture. Insulated covers and thin samples help. For other geometries and wider temperature ranges, gas convection is preferred. 

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

Steady state methods are usually based on parallel plate geometry, although coaxial cylinders are also suitable. The unguarded hot plate apparatus is a development of Lee s disk first described in 1898 and beloved of school physics laboratories for many decades. The general arrangement is shown in Figure 14.1. 

In the case of the thermal-conductivity, there are three main techniques those based on Equation (1) and those based on a transient application of it. Prior to about 1975, two forms of steady-state technique dominated the field parallel-plate devices, in which the temperature difference between two parallel disks either side of a fluid is measured when heat is generated in one plate, and concentric cylinder devices that apply the same technique in an obviously different geometry. In both cases, early work ignored the effects of convection. In more recent work, exemplified by the careful work in Amsterdam with parallel plates, and in Paris with concentric cylinders, the effects of convection have been investigated. Indeed, the parallel-plate cells employed in Amsterdam by van den Berg and his co-workers have the unique feature that, because the temperature difference imposed can be very small and the horizontal fluid layer very thin, it is possible to approach the critical point in a fluid or fluid mixture very closely (mK). 

Figure 2-9. A number of simple flow geometries, such as concentric cylinder (Couette), cone-and-plate, and parallel disk, are commonly employed as rheometers to subject a liquid to shear flows for measurement of the fluid viscosity (see, e.g., Fig. 3-5). In the present discussion, we approximately represent the flow in these devices as the flow between two plane boundaries as described in the text and sketched in this figure.

Figure 3-5. Typical rheometer geometries (a) parallel disk, (b) concentric cylinder (Couette) geometry, (c) cone-and-plate. Either the angular velocity is set and one measures the torque required to produce this rotation rate, or the torque is set and one measures the angular velocity. We analyze the Couette device in this section.

Since Oq is very small (ca. 0.5-3°, or 0.0087-0.0523 radians), sin (jt/2 -So) will close to one, and t wiU be nearly independent of position [see Eq. (59)] that is, the tested material between the gap will experience uniform shear stress. This is the advantage of cone-plate compared to other geometries (i.e., capillary tube and parallel disk). For example, at an angle of 1°, the pereentage difference in shear stress between cone and plate is 0.1218% (Fredrickson, 1964). This is within the precision of measurements that must be made therefore, one can assume that shear stress, and, hence, shear rate and apparent viscosity, are uniform throughout the fluid. 

L, the tip diffusion layer is not affected by the substrate and the current is ij oo and is independent of L. The standard type of tip is the cross section of a micron-sized wire sealed in an insulator such as glass that is beveled to allow close approach of the tip to the substrate. This tip geometry is most frequently employed because the tip surface is parallel to the substrate and maximizes the feedback effect discussed below. Conical electrodes such as etched STM-like tips are occasionally used when a disk geometry is not feasible, although the feedback (/t/ t,oo) is smaller [26]. 

Numerous variants of this technique have been created to adapt it for more practical applications. The most popular of these is the parallel-plate apparatus. Here the melt is placed between two parallel disks, 1-2 mm apart. The shear rate varies from the center to the circumference of the disk and corrections need to be performed [1]. The parallel-plate geometry is less sensitive to errors in gap and is also more able to handle filled materials. In contrast, cone-and-plate geometries are not useful in cases where the filler dimension is of the same order of magnitude as the gap. Parallel plates are also recommended in situations where rheology is measured as a function of temperature, where tool thermal expansion would otherwise affect the accuracy of the measurement. 

Three types of rotational rheometer are suitable for studying non-Newtonian fluids concentric cylinder, cone-plate and parallel disk. In each case, torque is measured as a function of angular velocity. Data analysis is simplest in the cone-plate geometry, because the shear rate is uniform throughout the sample and hence no correction is needed to obtain the true viscosity. The corresponding shear rate is... 

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