Flow exponent

Conductances (Imk types) Crack Flow coefficient, flow exponent... 

Flow exponent The exponent of the pressure difference in the flow equation. Its value ranges from 0.5 for turbulent flow to 1.0 for laminar flow. 

Fluid Factor K Flow exponent n Yield stress o0... 

Figure 3.3 illustrates the special cases of Eq. 3.6 used to describe Herschel-Bulkley fluids and, depending on the flow exponent and yield stress values, Newtonian fluids, shear thinning, shear thickening, and Bingham fluids. The values for Eq. 3.6 are given in Table 3.1. 

The residence time distribution also changes in the pipe in accordance with the flow profiles, depending on the rheological characteristics of the fluid with increasing flow exponent n, the width of the residence time distribution increases, i. e., for n — 0 the pipe flow practically is a plug flow with a very narrow residence time spectrum, while n —> °° results in a broad residence time distribution. 

If the material to be processed is subject to shear thinning, the linear relationships for the pressure and energy behavior illustrated above no longer apply. With shear thinning, there is a non-linear relationship between the shear rate and shear stress that is reflected in the flow curve (see Chapter 3). As a rule, the zero viscosity and one or two rheological time constants are enough to describe the flow curve with sufficient accuracy. The Carreau equation is often used it contains a dimensionless flow exponent in addition to the zero viscosity and a rheological time constant. 

The dimensionless flow exponent m and the rheological time constant are additional influencing variables that turn the one-dimensional problem for Newtonian fluids into a three-dimensional problem. The rheological time constant , when multiplied by the revolution speed n, forms an independent dimensionless group (Deborah number). 

The flow exponent m, which generally assumes values between 0.5 and ] in practice, is regarded as constant in the following, making the problem two-dimensional. If test results that were obtained with a model material are transferred to a real material system, the results will only apply to materials with the same flow exponents. Here we can see that model theory is limited in its usefulness. With complicated material behaviors, the amount of experimentation required increases vastly. The only solution is to use a material law that contains just one time constant as a parameter in addition to zero viscosity. Although suitable material laws do exist, they often provide an inaccurate description of the flow curve. 

One advantage of the concept of representative viscosity is that the measured flow curve is entirely sufficient to determine the pressure and energy behaviors. It is not necessary to determine the flow exponent m and the time constant . As can be seen in Fig. 7.12, the representative viscosity is determined with reference to the representative shear rate from the flow curve using Eq. 7.15. This then replaces the constant Newtonian viscosity in Eq. 7.2. 

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. 

As the flow exponent becomes smaller, the polymer melt becomes increasingly shear thinning, and the pressure required to achieve a specific flow rate becomes even smaller (Fig. 8.10). We also note that the intersection of the curve with the x-axis is not dependent on the flow exponent. There is no axial pressure gradient at the intersection. A simple two-dimensional analogy model is the planar channel. If only the walls are moved and no pressure gradient is overlaid, the flow rate, according to an analytical solution, is not dependent on the flow exponents. 

The power characteristic is also strongly dependent on the flow exponents (Fig. 8.11). The more shear thinning the polymer, the less power is required to convey the polymer melt. Unlike the conveying characteristic, the intersections of the individual curves with the x-axis lie far apart and are located outside the figure. 

Process engineers generally find simplified diagrams and formulas sufficient to estimate the pressure in these components. There is standardized pressure for melt filters, determined experimentally, which is dependent only on the screen mesh count and the flow exponent of the polymer. Should the screen mesh count be changed while all other parameters remain constant, the pressure loss of the screen will alter in proportion to this standardized pressure (Fig. 11.12). 

The effect of changed operating conditions or hole geometry on the pressure loss P at the discharge holes of the die plate can also be easily estimated with the help of the flow exponent m. The flow exponent for standard polymers is generally found to be between 2.5 and 3.5 (Fig. 11.13). 

Ostwald 1925) with a temperature dependant fluidity 0 and a flow exponent m as a measure of shear thinning. Ceramic materials with Newtonian behaviour above the stress threshold, Eqn. (5), are called Bingham materials, materials which exhibit shear thinning, Eqn. (6), are called Casson... 

The disperse crosslinked elastomer phase shows behavior similar to that of a polymeric filler. Viscosity is highly dependent on shear rate [7], but hardly dependent on temperature [7]. The flow exponent is between 3 and 5. 

With plastic melts, the flow exponent n, as it is called, lies between 2 and 3. Equation (7-58) can, of course, only apply to a limited range of shear stresses. 

To begin with, the reference values as set out above were used. When the delivery gap width B and the flow exponent n are varied, with a constant delivery gap height H, the pressure drop function shown in Fig. 2 is obtained, assuming an identical melt delivery rate in the model version and the main version. This diagram can be described by Equation 8. 

If the channel height and flow exponent are set as variable and the channel width kept constant, the scale-up rules shown in Fig. 3 and Equation 13 are obtained. Equations 13 and 14 describe the pressure loss for deviations from the channel reference values. 

Allowance can also be made for a change of material during the design process the scale-up rules achieve this through a variable flow exponent n. 

Fig. 2 Dimensionless pressure drop over dimensionless channel width and flow exponent... 

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