VISCOMETRY AND TUBE FLOW

The quantities ij, t, and y are defined in tenns of viscometric flows. This chapter considers the principles behind the devices used to establish viscometric flows and thereby determine the viscous flow properties discussed in the previous chapter. Details of the instruments and procedures have been reviewed. One of the common techniques, Poisueille (laminar) flow in cylindrical tubes, is also important from a technological standpoint, since polymer melts and solutions are often transported and processed in this fashion. 

The chapter concludes with a discussion of the turbulent flow of non-Newtonian polymer fluids. Because of the tremendous viscosities of polymer fluids, laminar flow is far more prevalent than in nonpolymer fluids in fact, turbulence is normally encountered only with dilute polymer solutions (less than a few weight percent, or so). Nevertheless, when turbulent flow does occur, there are some interesting and practically important difierences from the turbulent flow of Newtonian fluids. 

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The formulas just developed allow the relation of pressure gradient, velocity profile, and volumetric flow rate os long as the shear stress-shear rate relation (flow curve) for the fluid is known. The problem in viscometry is just the reverse How is the flow curve obtained from pressure drop-volumetric flow rate measurements in a cylindrical tube If the mathematical form of the flow curve, that is, a particular constitutive equation, is assumed a priori, the integrated equations as developed above may be used to establish the parameters in the constitutive relation. For example, if it is assumed that the power law represents the flow curve of a fluid under investigation, two readings of dP/dx versus Q in a tube of known R will allow calculation of K and n from (16.13). However, in the general case, the form of the constitutive equation is not known a priori, and must be established by viscometry. This may be done, first by integrating (16.12) by parts... 

Capillary viscometry measures the time it takes for the test fluid to flow through a capillary tube. The viscosity is calculated from this time and the calibration constant of the tube. 

The viscous dissipation term is normally not important. Its significance has been considered in connection with lubrication theory (VI), flow through tubes (B20), extrusion of plastics melts (BIO), and viscometry in rotating-cylinder systems (W6). There is also an additional contribution to the energy flux vector describing energy transport by radiation. See discussion in connection with Eq. (29). 

One of the simplest methods of examining this effect is by capillary viscometry, although automatic viscometers are commercially available. In a U-tube viscometer such as the Ubbelohde suspended level dilution model shown in Figure 9.8, the flow times of pure solvent and a polymer solution t are recorded. This is done by pipetting an aliquot of solution of known volume into bulb D. The solution is then pumped into E. The flow time t is the time taken for the solution meniscus to pass from X to y in bulb E. 

Rheology of Foams—Flow in a Tube. Rheological behavior of foams has been measured in a viscometry apparatus (Fig. 5.102). 138.145 Typically, the foam constituents are mixed and passed through a foam generator, such as a packed bed. The foam is then displaced through a small-diameter tube, and flow rates, pressure drop, and temperature are measured. 

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