The Power-Law Equation

In many situations, t]o rioc Ky l,and 77,is small. Then the Cross equation (with a simple change of the variables K and m) reduces to the well-known power-law (or Ostwald-de Waele) model, which is given by 

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Here /, r and, v are unequal integers in the set 1, 2, 3. As already mentioned, in the thin-layer approach the fluid is assumed to be non-elastic and hence the stress tensor here is given in ternis of the rate of deforaiation tensor as r(p) = riD(ij), where, in the present analysis, viscosity p is defined using the power law equation. The model equations are non-dimensionalized using... 

Another complication arises when not all of the internal surface of a porous catalyst is accessed. Then a factor called the effectiveness T is apphed, making the power law equation, for instance,... 

Since the power law equations are going to be used rather than the flow curves, it is necessary to use the true shear rates. 

We attempted to describe the integral dependences in Fig. 7 also by the power-law equations... 

Rate constant. It is easiest to estimate the value of the rate constant if it is the only unknown in the kinetic expression. Typically these expressions are the power law equations ... 

In general it is fair to say that rheologists have been conservative in their use of non-exponential kernels. One particular form clearly stands out as a candidate for describing experimental data, at least for a limited range of relaxation times. This is the power law equation, often applied to... 

The term S represents the strength of the network. The power law exponent m was found to depend on the stochiometric ratio r of crosslinker to sites. When they were in balance, i.e. r = 1, then m - 1/2. From Equations (5.140) and (5.141) this is the only condition where G (co) = G (cd) over all frequencies where the power law equation applies. If the stochiometry was varied the gel point was frequency dependent. This was also found to be the case for poly(urethane) networks. A microstructural origin has been suggested by both Cates and Muthumkumar38 in terms of a fractal cluster with dimension D (Section 6.3.5). The complex viscosity was found to depend as ... 

Work by the USBuMines in ground motion transmission (Ref 2) produced two particularly significant results for the typical charge weights and distances found in surface blasting 1) the constant m is equal to one-half of the constant n the power law equation then has only two unknowns and assumes the following form ... 

Fuvpi, %uvP2/ and Vuvp3 are the average liquid velocities for transducers 1, 2, and 3, respectively, from channel 0 to the channel where the gas-liquid interface is located. The constant 0.7 is obtained in the region 0single-phase turbulent flow the assumption made here is that the gas phase is located in the upper part of the pipe and the liquid velocity, not disturbed by the gas phase, develops in the lower part of the pipe as it does in single-phase turbulent flow. 

Figure 16c is a sample flow map of the liquid velocity profile 221. The gas phase occupies more than 50% of the cross-sectional area of the pipe, and it is not symmetrically distributed above the liquid phase. Figure 16c also shows a higher liquid velocity near the center of the pipe that decreases radially. The lighter (yellow) lines in the lower part of the pipe and near the wall correspond to the liquid velocity values obtained from the power law equation the darker lines (dark gray) above the power law equation values correspond to the near field effect of the transducer. 

The following comments can be made about the Power Law equation and the viscosity or flow curve, as, for example, that shown in Fig. 3.5 ... 

If the Power Law equation is used in pressure flows, where 0 < y < y,Tl. , an error is introduced in the very low shear rate Newtonian region. In flow rate computation, however, this is not a very significant (40). 

The slope of the viscosity curve in the Power Law region is not exactly constant. The flow index n decreases with increasing shear rate. Thus the Power Law equation holds exactly only for limited ranges of shear rate, for a given value of n. 

The polymer melt used in this example has a density of 1000 kg/m3. The following initially assumes a Newtonian flow behavior with a viscosity of 1000 Pa-s. In later computations, a more realistic shear thinning flow behavior is assumed, which can be described using the power law equation. The flow exponent n ranges between 0.4 and 0.9 and the consistency... 

The power law equation was used for the viscosity for the other curves in the two diagrams. The flow exponent n was varied between 0.4 and 0.9. The choice of the power law equation provides a non-linear relationship between the flow rate and the pressure and the flow rate and the power, respectively. We note that the flow exponent has significant influence both on the conveying characteristic and on the power characteristic. 

It is interesting to note that even the more realistic model adhering to the Case II radial and axial drug release from a cylinder, (4.10), can be described by the power-law equation. In this case, pure Case II drug transport and release is approximated (Table 4.1) by the following equation ... 

The nice fittings of the previous functions to the release data generated from (4.16) and (4.17), respectively, verify the argument that the power law can describe the entire set of release data following combined release mechanisms. In this context, the experimental data reported in Figures 4.8 to 4.10 and the nice fittings of the power-law equation to the entire set of these data can be reinterpreted as a combined release mechanism, i.e., Fickian diffusion and a Case II transport. 

The proportionality factor k is called the experimental rate constant with catalytic reactions, this constant is frequently a complex quantity which may be a product of rate constants of several steps or may include equilibrium constants of the fast steps. The exponents m,n,... in the power-law equations may be any fraction or small integer (positive, negative or zero). The constants K, in the denominator of the equations of type 7 are often but not always related to adsorption equilibrium constants. While they have to be evaluated from the experimental data, the values of exponents a, b, a, / and d arc derived from the assumed mechanism in the case of model rate equations. In empirical rate equations these constants can attain any value (fractional or small integer, usually positive) and... 

Percolation media can be characterized not only by the percolation probability but also by other quantities (Table II)—for example, by the correlation length, which is defined as the average distance between two sites belonging to the same cluster. Near the percolation threshold, all these quantities are usually assumed to be described by the power-law equations (Table II). All current available evidence strongly suggests that the critical exponents in these equations depend only on the dimensionality of the lattice rather than on the lattice structure (72). Also, bond and site percolations have the same exponents. 

For fluids that can be described by the power law (Equation 2.3), the generalized Reynolds number (GRe) can be calculated from the equation ... 

Power Law Fluid or Emulsion A fluid or emulsion whose rheological behavior is reasonably well-described by the power law equation. Here shear stress is set proportional to the shear rate raised to an exponent n, where n is the power law index. The fluid is pseudoplastic for n < 1, Newtonian for n = 1, and dilatant for n > 1. 

The maximum stress on the stress-strain curve in each experiment was plotted against the strain rate for the frozen sample in Figure 2. The relationship between the maximum stress (a) and the strain rate (e) can be expressed in the power law equation,... 

Measure the bulk viscosity of a polymer solution at different shear rates. Then we have J, versus y. Obtain K and n by fitting the data into the power-law equation ... 

Chen et al. (1998) used the power-law equation to describe the apparent viscosity in shear-thinning and shear-thickening regimes ... 

In some cases the very act of deforming a material can cause rearrangement of its microstructure such that the resistance to flow increases with an increase of shear rate. In other words, the viscosity increases with appHed shear rate and the flow curve can be fitted with the power law. Equation (20.3), but in this case n> 1. The shear thickening regime extends over only about a decade of shear rate. In almost all cases of shear thickening, there is a region of shear thinning at low shear rates. 

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