Power-function fluid

A. N. Prokunin and M. L. Fridman 23 24) seem to be the first investigators to have done the first adequately accurate calculations of power-function fluid helical flow in a circular (tubular) head with a rotary core. At the same time a series of experiments... 

Quantitative Analysis of Spiral Flow (Combined Shear) and Asymptotic Solutions for a Power-Function Fluid... 

The dimensionless relations are usually indicated in either of two forms, each yielding identical resiilts. The preferred form is that suggested by Colburn ran.s. Am. In.st. Chem. Eng., 29, 174—210 (1933)]. It relates, primarily, three dimensionless groups the Stanton number h/cQ, the Prandtl number c Jk, and the Reynolds number DG/[L. For more accurate correlation of data (at Reynolds number <10,000), two additional dimensionless groups are used ratio of length to diameter L/D and ratio of viscosity at wall (or surface) temperature to viscosity at bulk temperature. Colburn showed that the product of the Stanton number and the two-thirds power of the Prandtl number (and, in addition, power functions of L/D and for Reynolds number <10,000) is approximately equal to half of the Fanning friction fac tor//2. This produc t is called the Colburn j factor. Since the Colburn type of equation relates heat transfer and fluid friction, it has greater utility than other expressions for the heat-transfer coefficient. 

Since the process is more complex, the proposed method may not be valid for scale-up calculation. The combination of power and Reynolds number was the next step for correlating power and fluid-flow dimensionless number, which was to define power number as a function of the Reynolds number. In fact, the study by Rushton summarised various geometries of impellers, as his findings were plotted as dimensionless power input versus impeller... 

Fig. 4.2.2 Dimensionless shear rate and viscosity as a function of radius for a power-law fluid under the conditions shown in Figure 4.2.1. For a highly shear thinning material, the shear rate is large near the wall and close to zero near the center. The viscosity can vary by several orders of magnitude in the pipe.

The usual approach for non-Newtonian fluids is to start with known results for Newtonian fluids and modify them to account for the non-Newtonian properties. For example, the definition of the Reynolds number for a power law fluid can be obtained by replacing the viscosity in the Newtonian definition by an appropriate shear rate dependent viscosity function. If the characteristic shear rate for flow over a sphere is taken to be V/d, for example, then the power law viscosity function becomes... 

In a series of papers, Chhabra (1995), Tripathi et al. (1994), and Tripathi and Chhabra (1995) presented the results of numerical calculations for the drag on spheroidal particles in a power law fluid in terms of CD = fn(tVRe, ). Darby (1996) analyzed these results and showed that this function can be expressed in a form equivalent to the Dallavalle equation, which applies over the entire range of n and tVRe as given by Chhabra. This equation is... 

In natural waters, unattached microorganisms move with the bulk fluid [55], so that no flux enhancement will occur due to fluid motion for the uptake of typical (small) solutes by small, freely suspended microorganisms [25,27,35,41,56,57], On the other hand, swimming and sedimentation have been postulated to alleviate diffusive transport limitation for larger organisms. Indeed, in the planar case (large r0), the diffusion boundary layer, 8, has been shown to depend on advection and will vary with D according to a power function of Da (the value of a is between 0.3 and 0.7 [43,46,58]). For example, in Chapter 3, it was demonstrated that in the presence of a laminar flow parallel to a planar surface, the thickness of the diffusion boundary layer could be estimated by ... 

Several expressions of varying forms and complexity have been proposed(35,36) for the prediction of the drag on a sphere moving through a power-law fluid. These are based on a combination of numerical solutions of the equations of motion and extensive experimental results. In the absence of wall effects, dimensional analysis yields the following functional relationship between the variables for the interaction between a single isolated particle and a fluid ... 

For a power-law fluid, the viscosity is now a function of shear rate. Often times, a non- Newtonian viscosity is designated with the lowercase Greek letter eta, rj, to differentiate it from a Newtonian viscosity, jr ... 

To overcome this problem, the method of extrapolation has been proposed. It is found that in many cases, the unspecified function can be quite well represented by a power function. For example, for closed fluid-flow systems, the Froude group is negligible and thus... 

It can be shown (V2) that Eq. (51) is identical to an equation presented earlier by Alves, Boucher, and Pigford (A3) for power-function non-Newtonians. This is due to the fact that neglecting the derivative in Eq. (48) implies assumption of this type of fluid behavior. Eq. (51) is to be slightly preferred to that of Alves et al. on the basis that its readily possible to take the slope of a semilogarithmic plot of the term (1/n") — 1 versus the torque t in order to determine the error... 

Multi-functional power transmission fluid additive... 

The Strain Distribution Function of a Power Law Fluid, in Pressure Flow between Parallel Plates Consider two infinitely wide parallel plates of length L gap II. Polymer melt is continuously pumped in the x direction. Assuming isothermal steady, fully developed flow, (a) show that F(c) is given by... 

Fig. 12.45 The function F(n, ft) for the flow of Power Law fluids in an annular region. [Reprinted by permission from A. G. Fredrickson and R. B. Bird, Non Newtonian Flow in Annuli, Ind. Eng.

Both polymeric and some biological reactors often contain non-Newtonian liquids in which viscosity is a function of shear rate. Basically, three types of non-Newtonian liquids are encountered power-law fluids, which consist of pseudoplastic and dilatant fluids viscoplastic (Bingham plastic) fluids and viscoelastic fluids with time-dependent viscosity. Viscoelastic fluids are encountered in bread dough and fluids containing long-chain polymers such as polyamide and polyacrylonitrite that exhibit coelastic flow behavior. These... 

Conventional stirred-tank polymeric reactors normally use turbine, propeller, blade, or anchor stirrers. Power consumption for a power-law fluid in such reactors can be expressed in a dimensionless form, Ne = Reynolds number based on the consistency coefficient for the power-law fluid. Various forms for the function f(m) in terms of the power-law index have been proposed. Unlike that for Newtonian fluid, the shear rate in the case of power-law fluid depends on the ratio dT/dt and the stirrer speed N. For anchor stirrers, the functionality g developed by Beckner and Smith (1962) is recommended. For aerated non-Newtonian fluids, the study of Hocker and Langer (1962) for turbine stirrers is recommended. For viscoelastic fluids, the works of Reher (1969) and Schummer (1970) should be useful. The mixing time for power-law fluids can also be correlated by the dimensionless relation NO = /(Reeff = Ndfpjpti ) (Tebel et aL 1986). 

The Rabinowitsch correction and the velocity profile are simple analytical functions of the power law exponent n. A schematic diagram of velocity profiles for power law fluids is shown in Figure 13.6. 

Correlations for mass transfer are conveniently divided into those for fluid - fluid interfaces and those for fluid - solid interfaces. Many of the correlations have the same general form. That is, the Sherwood or Stanton numbers containing the mass transfer coefficient are often expressed as a power function of the Schmidt number, the Reynolds number, and the Grashof number. 

Compare the Carreau viscosity as a function of shear rate (Fig. 10.8 and Problem 10.3) with that of the power-law fluid in Problem 10.2. 

By substituting this function into (6.2.8)—(6.2.11), we can find the basic characteristics of the film flow of a power-law fluid along an inclined plane. The corresponding results are presented in Table 6.4. 

Pseudoplastic, in which the shear stress depends on the shear rate alone. Power law, in which the shear stress is not a linear but an exponential function of shear rate. The rheological expression for a power-law fluid is... 

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