Newtonian fluid flow index

A Newtonian liquid of viscosity 0.1 N s/m2 is flowing through a pipe of 25 mm diameter and 20 m in lenglh, and the pressure drop is 105 N/m2. As a result of a process change a small quantity of polymer is added to the liquid and this causes the liquid to exhibit non-Newtonian characteristics its rheology is described adequately by the power-law model and the flow index is 0.33. The apparent viscosity of the modified fluid is equal to ihc viscosity of the original liquid at a shear rate of 1000 s L... 

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. 

X is a function of the flow index n. For Newtonian fluids, the value of Xn is unity. 

The consistency index K (or K ), which characterizes the consistency or thickness of a fluid. It is analogous to the viscosity of a Newtonian fluid and similarly enables quantitative comparison of the consistency of fluids having identical flow-behavior indexes. 

As pointed out in Section I, viscosities are really meaningful if compared only for Newtonian fluids or at specified shear rates for other materials. A similar limitation must be imposed on the consistency indexes K and K values of either are comparable only for fluids with the same flow behavior indexes n or n . 

Decreases in concentration or increases in temperature usually decrease the consistency indexes K and K but leave the flow-behavior indexes n and n relatively unaltered. The latter appear to be determined primarily by the components of the non-Newtonian fluid and increase only slightly with increases in temperatures or decreases in concentration for pseudoplastic materials. 

Ward and DallaValle (W2) studied the two-phase cocurrent flow of air and four non-Newtonian fluids in three horizontal pipes (I.D. 0.82 to 1.60 in.). Flow rates were such that both phases were in turbulent motion. The flow-behavior index of the liquids used was varied from 0.31 to 1.00,... 

Values of the consistency index K and the flow behavior index n for a non-Newtonian fluid can be determined experimentally. For pseudoplastic fluids, n < 1. 

Figure 6.50 Cumulative residence time distributions for tube flow with a Newtonian fluid, and power law fluids with power law indeces of 0.5 and 0.1. Plug flow, which corresponds to a Bingham fluid with a power law index of 0 is also shown.

Therefore, when r -> Vo, the shear rate of a non-Newtonian fluid tends to that of a Newtonian fluid. In Table 3-1, values of the correction factor in parenthesis in Equation 3.7 are given for several values of the flow behavior index, n, and the ratio (n/ o) of the concentric cylinders, and they confirm that (1) they are small when n ro, and (2) they may be large when the fluids are substantially non-Newtonian and when (n/ro) < 0.95. 

A comprehensive example for sizing a pump and piping for a non-Newtonian fluid whose rheological behavior can be described by the Herschel-Bulkley model (Equation 2.5) was developed by Steffe and Morgan (1986) for the system shown in Figure 8-2 and it is summarized in the following. The Herschel-Bulkley parameters were yield stress = 157 Pa, flow behavior index = 0.45, consistency coefficient = 5.20 Pas". 

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 ... 

In Equation (2), n is the flow behavior index (-),K is the consistency index (Pa secn), and the other terms have been defined before. For shear-thinning fluids, the magnitude of nshear-thickening fluids n>l, and for Newtonian fluids n=l. For PFDs that exhibit yield stresses, models that contain either (Jo or a term related to it have been defined. These models include, the Bingham Plastic model (Equation 3), the Herschel-Bulkley model (Equation 4), the Casson model (Equation 5), and the Mizrahi-Berk model (Equation 6). 

The power-law index plays an important role in melt flow. It is obvious that the high flow rate results in the increase of the die pressure. However, for a Newtonian fluid (such as water, n = 1), a 10 X increase in pressure is accompanied by a 10 X increase in flow rate. For a non-Newtonian fluid with n = 1/2, a 10 X increase in pressure is accompanied by a 100 X increase in flow rate. For n = 1/3, a 10 X increase in pressure results is accompanied by a 1000 X increase in flow rate. For n = 1/4, a 10 X increase in pressure is accompanied by a 10,000 increase in flow rate. For n = 1/5, as in the case of a bad regrind above, a lOX increase in pressure results in a 100,000 increase in flow rate This is essentially applicable for any extrusion die. Hence, the power-law index of a hot melt essentially determines its extrusion behavior. 

Rotational speed, r/s n, critical speed for complete solids suspension Flow behavior index of non-newtonian fluid... 

Flow consistency index of non-newtonian fluid Thermal conductivity, W/m-°C or Btu/ft-h-°F kj, at mean film temperature k, of tube wall Length, m or ft... 

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