Power number INDEX

Yooi24) has proposed a simple modification to the Blasius equation for turbulent flow in a pipe, which gives values of the friction factor accurate to within about 10 per cent. The friction factor is expressed in terms of the Metzner and Reed(I8) generalised Reynolds number ReMR and the power-law index n. 

The Tj, factors correct for the non-Newtonian shear rheology effects that occur in the channel. The parameters that are used in the correction correlation include rheological and geometric factors power law index (n), aspect ratio of the channel [H/W], the ratio of the channel width to the screw diameter (W/Df), and the number of flight starts (p). 

When a number is repeatedly multiplied by itself in an arithmetic expression, such as 3 x 3 x 3, or A x A x A x A, the power or Index notation (also often called the exponent) is used to write such products in the forms 33 and, respectively. Both numbers are in the general form an, where n is the index. If the index, n, is a positive integer, we define the number a " as a raised to the th power. 

Figure 6.48 presents the reduction of the striation thickness as a function of number of revolutions, as well as ratio of inner to outer cylinder radius, k for a Newtonian fluid. If we were to plot the striation thickness for a shear thinning fluid, say a power law index n = 0.5, the naked eye would not be able to distinguish between the Newtonian and the shear thinning results. 

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). 

For normal mixing, the Pfaudler agitation-index (y) number for this low-viscosity fluid is 2 ft2/s3. Most stirrers are designed for impeller Reynolds numbers of 1000 or greater. For the impeller specified, the power number tjfn is 0.6 at high Reynolds numbers [2]. 

Exponential notation is an alternative way of expressing numbers in the form fl ( a to the power ), where a is multiplied by itself n times. The number a is called the base and the number n the exponent (or power or index). The exponent need not be a whole number, and it can be negative if the number being expressed is less than 1. See Table 39.2 for other mathematical relationships involving exponents. 

Dodge and Metzner (16) presented an extensive theoretical and experimental study on the turbulent flow of non-Newtonian fluids in smooth pipes. They extended von Karman s (17) work on turbulent flow friction factors to include the power law non-Newtonian fluids. The following implicit expression for the friction factor was derived in terms of the Metzner-Reed modified Reynolds number and the power law index ... 

TABLE 17.4 The power-law index, shear viscosity, and other useful properties of a number of neat generic polymers [14]... 

Figure 10.8 presents the fully established Nusselt values for the T boundary condition (i.e., constant temperature on all four walls) as a function of the power-law index n with the aspect ratio a as a parameter. Many predictions are shown for the plane parallel plates case (a = 0) covering the range of n values from 0 to 3. The corresponding Nusselt number decreases rather rapidly from a value of 9.87 at n = 0 to 7.94 at n = 0.5, then decreases more slowly to a value of 7.54 at n = 1.0. 

Turning next to the HI boundary condition, Fig. 10.9 presents the fully developed Nusselt number predictions for the plane parallel plates case covering the power-law index range from 0 to 3. The available predictions for the square duct, with n varying from 0.5 to 1, are also shown. As in the case of the T boundary condition, the slug flow and newtonian flow limits are also available for the HI condition for all aspect ratios. As in the constant-temperature case, a large decrease in the Nusselt number occurs for any aspect ratio when n increases from 0 to 0.5, and the subsequent decrease from 0.5 to 1.0 is more gentle. The dashed lines represent estimates of the fully established Nusselt values for intermediate values of the aspect ratio and power-law index. 

FIGURE 10.10 Fully established laminar Nusselt numbers for the H2 boundary condition as a function of the power-law index for plane parallel plates and for a square channel. 

The mathematician Sylvester [53] investigated the conditions for the existence of chemical graphs and concluded that if the difference between every two letters of an algebraically existent graph be raised to the power whose index is the number of bonds connecting them the permutation sum of the product of those powers most not vanish. A reproduction of some of the chemicogr hs studied by Sylvester are shown in Figure 12. 

Consider a non-Newtonian fluid with power-law index n and consistency index m. Construct appropriate dimensionless representations for the Reynolds, Schmidt, and mass transfer Peclet numbers. 

A simple dimensional analysis (see example 5.1) of this flow situation shows that the drag coefl cient can be expressed in terms of the Reynolds number and the power-law index, i.e. 

At low Nahme numbers the melting rate increases with barrel temperature. However, at high values of the Nahme number the melting rate reduces with increasing barrel temperature. There is a critical Nahme number above which increasing barrel temperature results in reduced melting rate. At a power law index of n = 0.5 the critical Nahme number is about 4.5. The critical Nahme number increases with the power law index as shown in Fig. 7.43. 

Critical Nahme number versus power law index... 

Equation 7.356 shows that the value of constant A depends on the Nahme number NNa, the power law index n, and wall temperatures 0,5 and 0,. ... 

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