Models for Non-Newtonian Flow

Performing numerical simulations of the extrusion process requires that the shear viscosity be available as a function of shear rate and temperature over the operating conditions of the process. Many models have been developed, and the best model for a particular application will depend on the rheological response of the resin and the operating conditions of the process. In other words, the model must provide an acceptable viscosity for the shear rates and temperatures of the process. The simple models presented here include the power law. Cross, and Carreau models. An excellent description of a broad range of models was presented previously by Tadmor and Gogos [4]. 

The power law viscosity model was developed by Ostwald [28] and de Waele [29]. The model has been used in the previous sections of this chapter and it has the form shown in Eq. 3.66. The model works well for resins and processes where the shear rate range of interest is in the shear-thinning domain and the log(ri) is linear with the log (7 ). Standard linear regression analysis is often used to relate the log 

The disadvantage of the power law model is that it cannot predict the viscosity in the zero-shear viscosity plateau. When the zero-shear viscosity plateau is included, a nonlinear model must be specified with additional fitting parameters. A convenient model that includes the zero-shear viscosity and utilizes an additional parameter is the Cross model [30]  

In many extrusion simulation calculations it is often adequate to approximate the polymer viscosity data using two straight-line functions. One line describes the 

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For non-Newtonian fluids the best that one can do at the present time is to make use of various empirical models for non-Newtonian flow. For example, for the incompressible Bingham plastic one can use in place of Eq. (28) the expression... 

The most effective model for non-Newtonian flow is the Ostwald-de Waele two-parameter, power-law fluid model [13]. The popularity of this model is easily traced to its Iractability in mathematical manipulations. In the power-law model, the apparent viscosity y = j. l CM Idy = r/y of a polymer fluid subject to a simple steady shear flow is given by... 

It is evident that application of Green s theorem cannot eliminate second-order derivatives of the shape functions in the set of working equations of the least-sc[uares scheme. Therefore, direct application of these equations should, in general, be in conjunction with C continuous Hermite elements (Petera and Nassehi, 1993 Petera and Pittman, 1994). However, various techniques are available that make the use of elements in these schemes possible. For example, Bell and Surana (1994) developed a method in which the flow model equations are cast into a set of auxiliary first-order differentia] equations. They used this approach to construct a least-sciuares scheme for non-Newtonian flow equations based on equal-order C° continuous, p-version hierarchical elements. 

In Eq. 6.6-16 the first term on the right-hand side is the drag flow and the second term is the pressure flow. The net flow rate is their linear superposition, as in the case of the Newtonian model in single screw extrusion. The reason that in this case this is valid for non-Newtonian flow as well is because the drag flow is simply plug flow. 

For non-Newtonian flow (based on a power law rheological model) ... 

Wu B. Computational fluid dynamics investigation of turbulence models for non-Newtonian fluid flow in anaerobic digesters. Environ Sci Technol 2010 44(23) 8989-95. 

The main requirements for a network model of non-Newtonian flow are as follows ... 

Based on capillary bundle model for non-Newtonian fluid flow ... 

The following mathematical model is proposed for non-Newtonian flow ... 

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. Hence, formulation of mathematical models for these processes can be based on the equations of non-Newtonian fluid mechanics. The general equations of non-Newtonian fluid mechanics provide expressions in terms of velocity, pressure, stress, rate of strain and temperature in a flow domain. These equations are derived on the basis of physical laws and... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

In order to overcome the shortcomings of the power-law model, several alternative forms of equation between shear rate and shear stress have been proposed. These are all more complex involving three or more parameters. Reference should be made to specialist works on non-Newtonian flow 14-171 for details of these Constitutive Equations. 

You must determine the horsepower required to pump a coal slurry through an 18 in. diameter pipeline, 300 mi long, at a rate of 5 million tons/yr. The slurry can be described by the Bingham plastic model, with a yield stress of 75 dyn/cm2, a limiting viscosity of 40 cP, and a density of 1.4 g/cm3. For non-Newtonian fluids, the flow is not sensitive to the wall roughness. 

Corresponding expressions for the friction loss in laminar and turbulent flow for non-Newtonian fluids in pipes, for the two simplest (two-parameter) models—the power law and Bingham plastic—can be evaluated in a similar manner. The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, extreme care should be exercised in its application, because any application involving extrapolation beyond the range of shear stress (or shear rate) represented by the data used to determine the model parameters can lead to misleading or erroneous results. 

This equation too is solved with the same boundary conditions as Eq. (148). A series of equations results when different combinations of fluids are used. There is no change for the first stage. All the terms of equation of motion remain the same except the force terms arising out of dispersed-phase and continuous-phase viscosities. The main information required for formulating the equations is the drag during the non-Newtonian flow around a sphere, which is available for a number of non-Newtonian models (A3, C6, FI, SI 3, SI 4, T2, W2). Drop formation in fluids of most of the non-Newtonian models still remains to be studied, so that whether the types of equations mentioned above can be applied to all the situations cannot now be determined. 

Two protocols are presented for non-Newtonian fluids. Basic Protocol 1 is for time-independent non-Newtonian fluids and is a ramped type of test that is suitable for time-independent materials. The test is a nonequilibrium linear procedure, referred to as a ramped or stepped flow test. A nonquantitative value for apparent yield stress is generated with this type of protocol, and any model fitting should be done with linear models (e.g., Newtonian, Herschel-Bulkley unithit). 

The above approach can be used to model non-Newtonian flows with a relatively high degree of accuracy. For example, Fig. 11.7 presents a comparison between a RFM solution... 

Non-Newtonian Flow between Jointly Moving Parallel Plates (JMP) Configuration Derive the velocity profile for isothermal Power Law model fluid in JMP configuration. 

Fig. 13.18 Predicted velocity field showing fountain flow around the melt front region for non-Newtonian fiber suspension flow at about half the outer radius of the disk. The reference frame is moving with the average velocity of melt front, and the length of arrow is proportional to the magnitude of the velocity. The center corresponds to z/b = 0 and wall is z/b = 1, where z is the direction along the thickness and b is half-gap thickness. [Reprinted by permission from D. H. Chung and T. H. Kwon, Numerical Studies of Fiber Suspensions in an Axisymmetric Radial Diverging Flow The Effects of Modeling and Numerical Assumptions, J. Non-Newt. Fluid Mech., 107, 67-96 (2002).]...

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