Power law indices

The described algorithm may not yield a converged solution in particular for values of power law index less than 0.5. To ensure convergence, in the iteration cycle (h + 1) for updating of the nodal pressures, an initial value found by... 

The correction factor for converting apparent shear rates at the wall of a circular cylindrical capillary to true shear rates is (3n + l)/4n, where n is the power law index of the polymer melt being extruded. 

A slit die is designed on the assumption that the material is Newtonian, using apparent viscous properties derived from capillary rheometer measurements, at a particular wall shear stress, to calculate the volumetric flow rate through the slit for the same wall shear stress. Using the correction factors already derived, obtain an expression for the error involved in this procedure due to the melt being non-Newtonian. Also obtain an expression for the error in pressure drop calculated on the same basis. What is the magnitude of the error in each case for a typical power law index n = 0.377... 

Polyethylene at 170°C passes through the annular die shown, at a rate of 10 x 10 m /s. Using the flow curves provided and assuming the power law index n = 0.33 over the working section of the curves, calculate the total pressure drop through the die. Also estimate the dimensions of the extruded tube. 

Figure 2 Variation of power law index n) with the percent NBR content in NBR-CSPE blend.

Figure 20 Variation of power law index (/i) with the percent ACM in XNBR-ACM blends.

The effect of power-law index on the velocity profile is seen by plotting equation 3.134 for various values of n, as shown in Figure 3.39. 

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. 

In a series of experiments on the flow of flocculated kaolin suspensions in laboratory and industrial scale pipelines(26-27-2Sl, measurements of pressure drop were made as a function of flowrate. Results were obtained using a laboratory capillary-tube viscometer, and pipelines of 42 mm and 205 mm diameter arranged in a recirculating loop. The rheology of all of the suspensions was described by the power-law model with a power law index less than unity, that is they were all shear-thinning. The behaviour in the laminar region can be described by the equation ... 

A liquid w hose rheology can be represented by the power law model is flowing under streamline conditions through a pipe of 5 mm diameter. If the mean velocity of flow in I nt/s and the velocity at the pipe axis is 1.2 m/s, what is the value of the power law index n ... 

Two liquids of equal densities, the one Newtonian and the other a non-Newtonian power law fluid, flow at equal volumetric rates down two wide vertical surfaces of the same widths. The non-Newtonian fluid has a power law index of 0.5 and has the same apparent viscosity as the Newtonian fluid when its shear rate is 0,01 s-1. Show that, for equal surface velocities of the two fluids, the film thickness for the non-Newtonian fluid is 1.125 times that of the Newtonian fluid. 

A fluid which exhibits non-Newtonian behaviour is flowing in a pipe of diameter 70 mm and the pressure drop over a 2 m length of pipe is 4 x 104 N/m2. A pitot lube is used to measure the velocity profile over the cross-section. Confirm that the information given below is consistent with the laminar flow of a power-law fluid. Calculate the power-law index n and consistency coefficient K. 

FIGURE 35.40 Pressure buildup in the nip region between two calender rolls as a function of the power-law index. 

The latter form is required to reflect the fact that the direction of the shear stress must reverse when the shear rate is reversed, and to overcome objections such as y , and therefore r, having imaginary values when y is negative. The power n is known as the power law index or flow behaviour index, and K as the consistency coefficient. 

Velocity profiles for power law fluids showing the effect of the power law index, n... 

Calculate the theoretical power in watts for a 0.25 m diameter, six-blade flat blade turbine agitator rotating at N = 4 rev/s in a tank system with a power curve given in Figure 5.10. The liquid in the tank is shear thinning with an apparent dynamic viscosity dependent on the impeller speed N and given by the equation fia = 25(N)n 1 Pa s where the power law index n = and the liquid density p = 1000 kg/m3. 

There is an expression that does not truly fit either class of behaviour, for power law fluids which can be expressed in terms of stress, rate or apparent viscosity with relative ease. They can describe shear thickening or thinning depending upon the sign of the power law index n ... 

Fig. 6. Variation of the power law index p with surface roughness and Monin-Obukhov length L (a) z = 10 m (b) z = 30 m. From Huang (1979).

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