The Capillary Viscometer

The need to measure fluid properties at shear rates higher than those accessible with rotational viscometers arises because deformation rates can easily reach 10 -10 sec in polymer processing operations. To attain these high shear rates, we use flow through capillaries or slits and calculate the viscometric functions from a knowledge of the pressure drop-versus-flow rate relationship. 

To obtain the shear rate at the wall, we note that the volumetric flow rate Q 

Integrating by parts and knowing that r equals zero at the centerline and equals zero at the tube wall yields the following  

Changing the independent variable from r to through the use of Eq. (14.4.2) and noting that x depends uniquely on y and vice versa gives 

Using Leibnitz rule to differentiate Eq. (14.4.7) with respect to gives 

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Rheological Property Determination. The rheology of an emulsion is often an important factor in determining its stability. Any variation in droplet size distribution, degree of flocculation, or phase separation frequently results in viscosity changes. Since most emulsions are non-Newtonian, the cone-plate type device should be used to determine their viscosity rather than the capillary viscometer. 

Two main types of viscometers are suitable for the determination of the viscosity of a polymer melt The rotation viscometer (Couette viscometer, cone-plate viscometer) and the capillary viscometer or capillary extrusiometer. The latter are especially suitable for laboratory use since they are relatively easy to handle and are also applicable in the case of high shear rates. With the capillary extrusiometer the measure of fluidity is not expressed in terms of the melt viscosity q but as the amount of material extruded in a given time (10 min). The amount of ex-trudate per unit of time is called the melt index or melt flow index i (MFI). It is also necessary to specify the temperature and the shearing stress or load. Thus MFI/2 (190 °C)=9.2 g/10 min means that at 190 °C and 2 kg load, 9.2 g of poly-... 

Various methods are used to examine the viscosity characteristics of metallized gels. Two types that have received extensive application are the cone and plate viscometer and the capillary viscometer. Both instruments can measure rheological characteristics at high shear rates, and the former is useful for low shear rate measurements as well. 

Equation (20), known as the Poiseuille equation, provides the basis for the most common technique for measuring the viscosity of a liquid or a dilute colloidal system, namely, the capillary viscometer. 

It is impossible to read much of the literature on viscosity without coming across some reference to the equation of motion. In the area of fluid mechanics, this equation occupies a place like that of the Schrodinger equation in quantum mechanics. Like its counterpart, the equation of motion is a complicated partial differential equation, the analysis of which is a matter for fluid dynamicists. Our purpose in this section is not to solve the equation of motion for any problem, but merely to introduce the physics of the relationship. Actually, both the concentric-cylinder and the capillary viscometers that we have already discussed are analyzed by the equation of motion, so we have already worked with this result without explicitly recognizing it. The equation of motion does in a general way what we did in a concrete way in the discussions above, namely, describe the velocity of a fluid element within a flowing fluid as a function of location in the fluid. The equation of motion allows this to be considered as a function of both location and time and is thus useful in nonstationary-state problems as well. 

To obtain a larger range in viscosities determined with a capillary viscometer, polymers from different batches were used to prepare the emulsions. The results obtained with the capillary viscometer are given in Figure 3. The ratio between the viscosities of the two components of the emulsions is about 10. 

The capillary viscometer. The most common and simplest device for measuring viscosity is the capillary viscometer. Its main component is a straight tube or capillary, and it was first used to measure the viscosity of water by Hagen [28] and Poiseuille [60], A capillary rheometer has a pressure driven flow for which the velocity gradient or strain rate and also the shear rate will be maximum at the wall and zero at the center of the flow, making it a non-homogeneous flow. 

For non-Newtonian liquids the capillary viscometer is inappropriate, although in principle capillary viscometers of different internal radii could be used to give data for different average shear rates. A very wide range of such averages would be needed. In practice a concentric cylinder or related rheometer is used instead. 

In practice, we can measure viscosity by the efflux time of a liquid in a viscometer (Fig. 47.3). The capillary viscometer is made of two bulbs connected by a tube in which the liquid must flow through a capillary tube. The capillary tube provides a laminary flow in which concentric layers of the liquid slide past each other. Originally, the liquid is placed in the storage bulb (A). By applying suction above the capillary, the liquid is sucked up past the upper calibration mark. With a stopwatch in hand, the suction is released and the liquid is allowed to flow under the force of gravity. The timing starts when the meniscus of the liquid hits the upper calibration mark. The timing ends when the meniscus of the liquid hits the lower calibration mark of the viscometer. The time elapsed between these two marks is the efflux time. 

Although the values of Intrinsic viscosity determined with a low shear viscometer are the only ones which truly represent the Intrinsic viscosity at high molecular weights, the results from the capillary viscometer are shown In Figure 7 to give an Indication of the effect of shear In the viscosity range of the study. The values of Intrinsic viscosity are different for the two types of viscometers, but the trend of Intrinsic viscosity versus conversion Is still the same. 

All viscosities (except those in the temperature study) were measured at 77°F with the capillary viscometer and they were corrected to 10 sec l if not noted otherwise. The capillary viscometer has an accuracy of 0.1% for a 10 cp fluid at 10 sec l. 

In the capillary viscometer, the slip phenomena can also be observed due to inhomogenous flow (Cohen and Metzner, 1986 de Vargas et ai, 1993). Therefore, the influence of the slip phenomenon must be considered when analyzing the data. In the case of 0.2% xanthan gum solution, the slip velocity is an increasing function of the wall shear stress and also of the length to diameter ratio UD. However, the slip velocity becomes independent of UD at large UD (de Vargas et ai, 1993). 

The capillary viscometers shown in Figure 9-21 differ in their areas of application. Because of their low price and the fact that they only require low amounts, 3 cm, of liquid. Ostwald viscometers are by far the most frequently used. The amount of liquid used must be measured in very exactly and maintained constant otherwise the pressure head will vary with the different solutions. 

It should, however, be noted that there exist some hints that bicontinuous microemulsions behave elastically. This has been assumed to be due to the differences observed in measuring viscosities once in a Couette flow and in the other case by a capillary viscometer. Here it was observed that the values obtained with the capillary viscometer are markedly higher. It has been suggested that in capillary flow a component of elongational flow is observed and that in this type of flow elastic components can be observed much earlier than in shear flow [106,107]. 

Both the capillary viscometer (providing about 0.7% accuracy), the theory of which is based on the Hagen-Poiseuille equation and the oscillating disc viscometer (providing about 0.2% accuracy) are applicable to experimental determination of viscosity at high pressures and temperatures. 

Pressure-driven devices include capillary viscometers and slit-die viscometers, in both of which the flow is driven by pressure. In the case of the capillary viscometer the pressure is generated by an upstream piston, and in the case of the slit-die viscometer flow is generated by an extruder. In both cases, measurements of pressure drop and flow rate are used to determine the viscosity. Both techniques have the inherent problem of pressure drop, which may result in phase separation. For this reason, the techniques are suitable for low-pressure measurements, which may mean that the polymer has not reached equilibrated CO2 concentrations. 

The proper design of a profile modification treatment requires measurement of the permeability reduction caused by a given gel system. Core tests have shown that relative gel strength measurements with the capillary viscometer correlate with permeability reduction--Increasing gel strength develops higher residual resistance factors. 

For the capillary viscometer, L and R remain constant. Therefore, for any two measurements, Equation (2) becomes... 

But, for the capillary viscometer Q is proportional to 1/FT, and pressure drop is proportional to V where FT = flowing time in seconds and V is the applied vacuum in inches of mercury. 

In the capillary viscometer shown in Figure 6.1, if we denote by t the time required for a volume of fluid V to flow through the capillary, and take into consideration the variation of the geometrical size of the capillary, and fliat of mercury and sample densities, then Equation (6.4) becomes... 

The pressure difference can be generated by a liquid pump, but, in the capillary viscometer in the vertical position, it is the gravity that causes AP, which is given by... 

The flow behavior of non-Newtonian fluids is usually described by expressing either shear rate or viscosity as a function of shear stress. Absolute viscometers, either capillary or rotational, are used to perform the necessary measurements. In the capillary viscometer, the flow rate is measured as a function of applied pressure. Apparent viscosities calculated by means of Poiseuille s relation [Eq. (9)], are converted to true viscosities using the Weissen-berg-Rabinowitsch correction... 

Here aj = M/IttR L is the stress at the inner cylinder and N is the slope of a log-log graph of O vs. M. The corresponding correction for the parallel-disk rheometer is known as the Burgers correction, and is similar to the Weissenberg-Rabinowitsch correction described above for the capillary viscometer ... 

For example, in Chapter 6, to begin with three parameters, p (shear stress), e (shear strain), and E (modulus or rigidity), are introduced to define viscosity and viscoelasticity. With respect to viscosity, after the definition of Newtonian viscosity is given, a detailed description of the capillary viscometer to measure the quantity t follows. Theories that interpret viscosity behavior are then presented in three different categories. The first category is concerned with the treatment of experimental data. This includes the Mark-Houwink equation, which is used to calculate the molecular weight, the Flory-Fox equation, which is used to estimate thermodynamic quantities, and the Stockmayer-Fixman equation, which is used to... 

Viscosity of liquid Freon-22 on the saturation line was measured by Gordon and coworkers [0.40] utilizing the capillary-viscometer method with a floating level. The data in [0.40] lie 3-7% lower than the results of Phillips and Murphy [1.50], obtained using the same method. The discrepancy in the experimental results obtained using the rolling-ball method is even greater (Fig. 25). 

In most cases viscosity is measured by capillary viscometers or rotating viscometers. In a capillary viscometer one measures the pressure drop by means of constant laminar flow in a capillary the constant flow can be achieved by a pump and the pressure drop is obtained by a differential pressure transmitter whose plus and minus sides are connected to the capillary. The pressure drop is then directly proportional to the viscosity according to the Hagen-Poiseuille law [4, 11] [Eq. (30), where p is the viscosity, r is the capillary radius, I is the capillary length, Ap is the pressure drop, and is the mass flow rate]. The capillary viscometer may also be employed in-line for monitoring of molecular weight in polymerizations, as described in Ref. 14. 

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