Concentric cylinders shear rate

With several springs, which function as torque gauges, and a number of spindles, viscosities can be measured up to 10 mPa-s with the Brookfield viscometer. The shear rates depend on the model and the sensor system they are ca 0.1 100 for the disk spindles, <132 for concentric cylinders, and <1500 for the cone—plate forlow viscosity samples. Viscosities at very low (ca 10 — 1 )) shear rates can be measured with the concentric... 

The Ravenfield model BS viscometer is a wide shear rate range iastmment with several possible measurement systems cone—plate, parallel plates, concentric cylinders, and taper plug. The last gives shear rates of up to 10 , and the cone—plate of up to 8 x lO". The viscosity range is... 

In this apparatus the polymer melt is sheared between concentric cylinders. The torque required to rotate the inner cylinder over a range of speeds is recorded so that viscosity and strain rates may be calculated. 

Soule et al. [141] constructed a sparged, concentric cylinder bioreactor for the cultivation of suspensions of Pirus malus. Growth was reduced under all rotational conditions. Sun and Linden [106] employed a rotating wall vessel (Rotary Cell Culture System, Synthecon, Houston, TX, USA) to cultivate suspensions of Taxus cuspidata under laminar flow conditions. Shear rates were... 

Viscometric flows used for measurements include well known flows, such as flow in a narrow gap concentric cylinder device and between a small angle cone and a flat plate. In both of these cases the flows established in these devices approximate almost exactly simple shearing flow. There are other viscometric flows in which the shear rate is not constant throughout, these include the wide gap concentric cylinder flow and flow in a circular pipe, discussed above. 

As the name implies, the cup-and-bob viscometer consists of two concentric cylinders, the outer cup and the inner bob, with the test fluid in the annular gap (see Fig. 3-2). One cylinder (preferably the cup) is rotated at a fixed angular velocity ( 2). The force is transmitted to the sample, causing it to deform, and is then transferred by the fluid to the other cylinder (i.e., the bob). This force results in a torque (I) that can be measured by a torsion spring, for example. Thus, the known quantities are the radii of the inner bob (R ) and the outer cup (Ra), the length of surface in contact with the sample (L), and the measured angular velocity ( 2) and torque (I). From these quantities, we must determine the corresponding shear stress and shear rate to find the fluid viscosity. The shear stress is determined by a balance of moments on a cylindrical surface within the sample (at a distance r from the center), and the torsion spring ... 

Common geometries used to make viscosity measurements over a range of shear rates are Couette, concentric cylinder, or cup and bob systems. The gap between the two cylinders is usually small so that a constant shear rate can be assumed at all points in the gap. When the liquid is in laminar flow, any small element of the liquid moves along lines of constant velocity known as streamlines. The translational velocity of the element is the same as that of the streamline at its centre. There is of course a velocity difference across the element equal to the shear rate and this shearing action means that there is a rotational or vorticity component to the flow field which is numerically equal to the shear rate/2. The geometry is shown in Figure 1.7. 

When the shear rate reaches a critical value, secondary flows occur. In the concentric cylinder, a stable secondary flow is set up with a rotational axis perpendicular to both the shear gradient direction and the vorticity axis, i.e. a rotation occurs around a streamline. Thus a series of rolling toroidal flow patterns occur in the annulus of the Couette. This of course enhances the energy dissipation and we see an increase in the stress over what we might expect. The critical value of the angular velocity of the moving cylinder, Qc, gives the Taylor number ... 

The Mooney arrangement of a bob with a conical base is an attractive design as it is relatively easy to fill and uses the base area to enhance the measurement sensitivity. However the cone angle must be such that the shear rates in both the cone and plate and concentric cylinder sections are the same. This means that the gap between the cylinders must be very slightly larger than the gap at the edge of the cone and plate if a constant shear rate is required. Unfortunately the DIN standard bob is poor in this respect. 

The monodisperse materials described hereafter were obtained with the Couette type cell designed by Bibette et al. [ 150,159]. It consists of two concentric cylinders (rotor and stator) separated by a very narrow gap (100 pm), allowing application of spatially homogeneous shear rates over a very wide range (from 0 to 14280 s ), with shearing durations of the order of 10 s. 

EXAMPLE 4.2 Comparison Between Capillary Viscometers and Concentric-Cylinder Viscometers. Criticize or defend the following proposition A set of capillary viscometers with different radii can be used in much the same way as a concentric-cylinder viscometer with variable speed or gap width to conduct studies in which the rate of shear is an independent variable. 

Solution Equation (16) shows that the velocity gradient is not uniform in a capillary viscometer any more than it is in a concentric-cylinder instrument. The rate of shear dvldr is directly proportional to the radial distance from the axis of the cylinder. At the wall it has its maximum value, which is proportional to Rc] at the center of the tube it equals zero. Some intermediate value, say, the average, might be used to characterize the gradient in a given instrument. This quantity will be different for capillaries of different radii. All of this is similar to the situation in concentric-cylinder viscometers. 

This section describes common steps designed to measure the viscosity of non-Newtonian materials using rotational rheometers. The rheometer fixture that holds the sample is referred to as a geometry. The geometries of shear are the cone and plate, parallel plate, or concentric cylinders (Figure HI. 1.1). The viscosity may be measured as a function of shear stress or shear rate depending upon the type of rheometer used. 

In the first of these techniques an approximation to uniform rate of shear throughout the sample is achieved by shearing a thin film of the liquid between concentric cylinders. The outer cylinder can be rotated (or oscillated) at a constant rate and the shear stress measured in terms of the deflection of the inner cylinder, which is suspended by a torsion wire (Figure 9.2) or the inner cylinder can be rotated (or oscillated) with the outer cylinder stationary and the resistance offered to the motor measured. 

The Couette rheometer. Another rheometer commonly used in industry is the concentric cylinder or Couette flow rheometer schematically depicted in Fig. 2.48. The torque, T, and rotational speed, 0, can easily be measured. The torque is related to the shear stress that acts on the inner cylinder wall and the rate of deformation in that region is related to the rotational speed. The type of flow present in a Couette device is analyzed in detail in Chapter 5. 

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