Power-law exponent

Various relations have been proposed linking the various power law exponents for a homologous series under specified conditions [ 10,62] such as ... 

Fig. 8 The Haug triangle. The three extremes of conformation compact sphere, random coil and rigid rod) are placed at the apices of a triangle. The conformation of a given macromolecule is represented by a locus along the sides of the triangle between these extremes. Knowledge of the power law exponents (see text) can help to give us an idea of the conformation type. From [61]...

Here, the equation for the crack growth rate Aa/AN is defined in terms of the maximum energy release rate Tmax. the power-law exponent F R), and the point at which the crack growth rate data converge. The power-law exponent F R) for the Mars-Fatemi model is of the form... 

The y-velocities are all set to zero the problem is numerically underconstrained otherwise. Figure 2 also shows the finite-element prediction of this velocity profile for two cases a Newtonian fluid (power-law exponent = 1) and a shear-thinning fluid (power-law... 

Power law exponents r - and apparent activation energies as a fimetion of ... 

Many materials are conveyed within a process facility by means of pumping and flow in a circular pipe. From a conceptual standpoint, such a flow offers an excellent opportunity for rheological measurement. In pipe flow, the velocity profile for a fluid that shows shear thinning behavior deviates dramatically from that found for a Newtonian fluid, which is characterized by a single shear viscosity. This is easily illustrated for a power-law fluid, which is a simple model for shear thinning [1]. The relationship between the shear stress, a, and the shear rate, y, of such a fluid is characterized by two parameters, a power-law exponent, n, and a constant, m, through... 

Fig. 4.2.1 Dimensionless velocity profiles for steady flow in a circular pipe for a power-law fluid for values of the power-law exponent n = 0.25 (solid line), through to 0.5, 0.75 and 1 (small dashed line), respectively. As the power-law exponent decreases, the profile becomes progressively more blunt with most of the shearing being confined to the region near the wall.

The analysis may be simplified by postulating symmetry of the diverging 2max on both sides of the gel point [18]. A power law exponent (see Eq. 1-6) which is the same on both sides,... 

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 ... 

Pi is the steady state or final value of the coefficient of friction, and N is the corresponding maximum number of cycles after which there is no further degradation in p. n is a power law exponent and is positive. Other functions such as an exponential function or a reciprocal function can be used in place of the power law as far as they eliminate the boundary condition of p(A ) = Pi for the number of cycles greater than A i. However, any realistic friction degradation function should always be consistent with independent experimental measurements. 

In the in situ consolidation model of Liu [26], the Lee-Springer intimate contact model was modified to account for the effects of shear rate-dependent viscosity of the non-Newtonian matrix resin and included a contact model to estimate the size of the contact area between the roller and the composite. The authors also considered lateral expansion of the composite tow, which can lead to gaps and/or laps between adjacent tows. For constant temperature and loading conditions, their analysis can be integrated exactly to give the expression developed by Wang and Gutowski [27]. In fact, the expression for lateral expansion was used to fit tow compression data to determine the temperature dependent non-Newtonian viscosity and the power law exponent of the fiber-matrix mixture. 

Rotational speeds may be expressed either in terms of rpm or in terms of hertz. The power law exponent, n, has a definite physical significance. The value of n and the corresponding significance are determined either empirically or through theoretical means. Table 1 lists the most common values assigned to n. 

Table 1 Most Common Values Assigned to the Power Law Exponent, , When...

The basis of the scale-of-agitation approach is a geometric scale-up with the power law exponent, = 1 (Table 1). This provides for equal fluid velocities in both large- and small-scale equipment. Furthermore, several dimensionless groups are used to relate the fluid properties to the physical properties of the equipment being considered. In particular, bulk-fluid velocity comparisons are made around the largest blade in the system. This method is best suited for turbulent flow agitation in which tanks are assumed to be vertical cylinders. 

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