Cone and plate

MODELS BASED ON DECOUPLED FLOW EQUATIONS -SIMULATION OF THE FLOW INSIDE A CONE-AND-PLATE RHEOMETER... 

In Chapter 4 the development of axisymmetric models in which the radial and axial components of flow field variables remain constant in the circumferential direction is discussed. In situations where deviation from such a perfect symmetry is small it may still be possible to decouple components of the equation of motion and analyse the flow regime as a combination of one- and two-dimensional systems. To provide an illustrative example for this type of approximation, in this section we consider the modelling of the flow field inside a cone-and-plate viscometer. 

In the Couette flow inside a cone-and-plate viscometer the circumferential velocity at any given radial position is approximately a linear function of the vertical coordinate. Therefore the shear rate corresponding to this component is almost constant. The heat generation term in Equation (5.25) is hence nearly constant. Furthermore, in uniform Couette regime the convection term is also zero and all of the heat transfer is due to conduction. For very large conductivity coefficients the heat conduction will be very fast and the temperature profile will... 

Using the described algorithm the flow domain inside the cone-and-plate viscometer is simulated. Tn Figure 5.17 the predicted velocity field in the (r, z) plane (secondary flow regime) established inside a bi-conical rheometer for a non-Newtonian fluid is shown. 

Olagunju, D.O. and Cook, L. P., 1993. Secondary flows in cone and plate flow of an Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 46, 29-47. 

Petera, J. and Nassehi, V., 1995. Use of the finite element modelling technique for the improvement of viscometry results obtained by cone-and-plate rheometers. J. Non-Newtonian Fluid Mech. 58, 1-24. 

Normal Stress (Weissenberg Effect). Many viscoelastic fluids flow in a direction normal (perpendicular) to the direction of shear stress in steady-state shear (21,90). Examples of the effect include flour dough climbing up a beater, polymer solutions climbing up the inner cylinder in a concentric cylinder viscometer, and paints forcing apart the cone and plate of a cone—plate viscometer. The normal stress effect has been put to practical use in certain screwless extmders designed in a cone—plate or plate—plate configuration, where the polymer enters at the periphery and exits at the axis. 

Fig. 28. Cone—plate viscometer. R is the radius of the cone, a is the angle between cone and plate, and Q is the relative angular velocity.

Viscoelastic Measurement. A number of methods measure the various quantities that describe viscoelastic behavior. Some requite expensive commercial rheometers, others depend on custom-made research instmments, and a few requite only simple devices. Even quaHtative observations can be useful in the case of polymer melts, paints, and resins, where elasticity may indicate an inferior batch or unusable formulation. Eor example, the extmsion sweU of a material from a syringe can be observed with a microscope. The Weissenberg effect is seen in the separation of a cone and plate during viscosity measurements or the climbing of a resin up the stirrer shaft during polymerization or mixing. 

J The viscosity characteristics of a polymer melt are measured using both a capillary rheometer and a cone and plate viscometer at the same temperature. The capillary is 2.0 mm diameter and 32.0 mm long. For volumetric flow rates of 70 x 10 m /s and 200 x 10 m /s, the pressures measured just before the entry to the capillary are 3.9 MN/m and 5.7 MN/m, respectively. 

Assuming that the melt viscosity is a power law function of the rate of shear, calculate the percentage difference in the shear stresses given by the two methods of measurement at the rate of shear obtained in the cone and plate experiment. 

Now the cone and plate gives true shear rate whereas the ram extruder uses apparent shear rate. The Non-Newtonian correction factor is... 

Therefore the true shear rate on the cone and plate is equivalent to a shear rate of 0.69(1.18) = 0.817 on the ram extruder... 

Angle between cone and plate in viscometers or angle ... 

Data obtained wirh a cone and plate viscometer were ... 

Striking support of this contention is found in recent data of Castro (16) shown in Figure 14. In this experiment, the polymerization (60-156) has been carried out in a cone-and-plate viscometer (Rheometrics Mechanical Spectrometer) and viscosity of the reaction medium monitored continuously as a function of reaction time. As can be seen, the viscosity appears to become infinite at a reaction time corresponding to about 60% conversion. This suggests network formation, but the chemistry precludes non-linear polymerization. Also observed in the same conversion range is very striking transition of the reaction medium from clear to opaque. 

Models based on Eqs. (47)-(50) have been used in the past to describe the disruption of unicellular micro-organisms and mammalian (hybridoma) cells [62]. The extent of cell disruption was measured in terms of loss of cell viability and was found to be dependent on both the level of stress (deformation) and the time of exposure (Fig. 25). All of the experiments were carried out in a cone and plate viscometer under laminar flow conditions by adding dextran to the solution. A critical condition for the rupture of the walls was defined in terms of shear deformation given by Eq. (44). Using micromanipulation techniques data were provided for the critical forces necessary to burst the cells (see Fig. 4)... 

Fig. 25. Total and viable cell concentrations of TB/C3 hybridomas versus duration of shear in a cone and plate viscometer (shear stress 208 Nm ). The error bars indicate the 95% confidence intervals [62]...

Cone-and-plate viscometers have been employed to study shear effects in both suspended (e.g. [138]) and anchorage dependent [122] mammalian cells. These devices have the advantage of requiring only small sample volumes ( lml). However, they are generally inappropriate for plant cell suspensions due to the larger cell and aggregate sizes. 

Gel formation was monitored using a controlled-stress rheometer (Carri-Med CS 50, TA Instruments, Guyancourt, France) with cone-and-plate geometry (cone diameter 4 cm, angle 3°58 ). The bottom plate was fitted with a Peltier temperature controller that... 

Deviation from laminar shear flow [88,89],by calculating the material functions r =f( y),x12=f( Y),x11-x22=f( y),is assumed to be of a laminar type and this assumption is applied to Newtonian as well as viscoelastic fluids. Deviations from laminar flow conditions are often described as turbulent, as flow irregularities or flow instabilities. However, deviation from laminar flow conditions in cone-and-plate geometries have been observed and analysed for Newtonian and viscoelastic liquids in numerous investigations [90-95]. Theories have been derived for predicting the onset of the deviation of laminar flow between a cone and plate for Newtonian liquids [91-93] and in experiments reasonable agreements were found [95]. 

Flow irregularities at gap angles of 30° were observed in viscoelastic liquids [94]. It has been indicated in theoretical treatments that the possibility of secondary flows [96,97] in rotational devices is to be expected if the gap angle is much greater than 5°. For viscoelastic fluids deviations from laminar flow have only been reported in cone-and-plate geometries with gap angles above 10°. 

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